- Email: [email protected]
Instability of local deformations of an elastic rod
Physica D 182 (2003) 103–124
Instability of local deformations of an elastic rod S. Lafortune∗ , J. Lega Department of Mathematics, University of Arizona, P.O. Box 210089, Tucson, AZ 85721-0089, USA Received 31 May 2002; received in revised form 14 March 2003; accepted 26 March 2003 Communicated by C.K.R.T. Jones
Abstract We study the instability of pulse solutions of two coupled non-linear Klein–Gordon equations by means of Evans function techniques. The system of coupled Klein–Gordon equations considered here describes the near-threshold dynamics of a three-dimensional elastic rod with circular cross-section, subject to constant twist. We determine a condition on the speed of the traveling pulse which ensures spectral instability. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Elastic rod; Klein–Gordon equations; Evans function
1. Introduction When subject to sufficiently high constant twist, elastic rods kept under tension tend to assume a helical shape. Near this bifurcation threshold, the local deformation and twist of the filament can be described in terms of envelope equations for the slowly varying (complex) amplitude of the helical mode and slowly varying twist. Such envelope equations were derived by Goriely and Tabor [1] in the case of filaments with circular cross-section, and take the form of two coupled non-linear Klein–Gordon equations. Lega and Goriely gave a complete classification of traveling wave solutions to this system [2] (see also [3]) and considered the stability of some special solutions, such as traveling holes and stationary periodic structures [2]. A particular family of solutions found in [2] consists of traveling pulses, which correspond to a localized helical structure propagating at constant speed on an otherwise undeformed filament. A reconstruction of such a pulse solution is shown in the numerical simulation of Fig. 1. As illustrated in this figure, reproduced from [2], numerical studies indicate that some pulse solutions are able to propagate in a stable fashion. From the numerics, it is also clear that not all pulses are stable. The goal of this paper is to find a criterion on the speed of the pulse solution which ensures instability. More precisely, we identify a range of speeds for which the point spectrum of the linearized operator about a pulse solution has an eigenvalue on the positive real axis. To this end, we analyze the behavior of the Evans function for the linearized operator near the origin and at infinity, and use a parity argument to conclude that, in some parameter regime, this function vanishes on the positive real axis. ∗
Corresponding author. E-mail addresses: [email protected] (S. Lafortune), [email protected] (J. Lega). 0167-2789/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-2789(03)00125-8
104
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
Fig. 1. Numerical simulation (reproduced from [2]) of the non-linear Klein–Gordon equations (2.1), showing the stable propagation of a traveling pulse solution (given by Eqs. (2.2) and (2.4)). Each panel shows a reconstruction of the corresponding elastic filament. Parameters used in the simulation are ω = 0, µ = −864 and c02 = 2.4. The speed of the pulse is c 0.6124, and the box length is 1 dimensionless unit.
The Evans function [4–7] of a linear operator acting on functions defined on the real line is an analytic function on some region of the complex plane which vanishes on the point spectrum of the operator, and which takes on real values on the real axis. It can be expressed as a scattering coefficient [6], as a determinant whose entries involve inner products of an eigenfunction of the linear operator with an eigenfunction of its adjoint operator [4,5,8–10], or as a Wronskian [7,10–13]. Applications of this technique include stability results for the propagation of an action-potential in nerve axon equations [4] and in the FitzHugh–Nagumo equations [5–7], for pulse solutions to the generalized Korteweg–de Vries, Benjamin–Bona–Mahoney and Boussinesq equations [8], for multi-pulse solutions to reaction–diffusion equations [14,15], for solutions to perturbed non-linear Schrödinger equations [12,16,17] and to near integrable systems [18], and for traveling hole solutions of the one-dimensional complex Ginzburg–Landau equation near the non-linear-Schrödinger limit [19]. The point of view we adopt here is that reviewed by Sandstede [13], where the Evans function is defined as a Wronskian, which vanishes if initial conditions giving convergence at plus and minus infinity are linearly dependent. Because it is often difficult or even impossible to calculate the Evans function for all complex values of the spectral parameter, instability of a solution to a system of partial differential equations is typically proven by showing that the Evans function of the associated linearized operator has to vanish at least once on the positive real axis. More precisely, knowledge of the behavior of the Evans function near the origin and for real and large values of the spectral parameter are often sufficient to conclude that, by continuity, the Evans function should vanish for some real positive value of the spectral parameter. This argument is classical (see for instance [4–6,8,11,14,15,20,21]), and it is also what we use in this paper. The calculations presented here however differ from standard applications of Evans function techniques for the following reasons. First, the linear system we are interested in has three-dimensional stable and unstable eigenspaces at ±∞, which we have to track carefully as the spectral parameter goes from the origin to infinity. Second, we will see that the first four derivatives of the Evans function vanish at the origin. In order to calculate the fifth derivative, we look for expansions of the eigenfunctions of the linear operator in powers of the spectral parameter. But because the asymptotic linear operator has a marginal mode which is bounded but
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
105
non-decaying at infinity, both regular and singular expansions are needed. The existence of a zero of order five of the Evans function may seem surprising for a Hamiltonian system [22], but we will see that another particularity of this problem is that the eigenvalue zero is embedded in the continuous spectrum. This paper is organized as follows. The coupled Klein–Gordon equations and their one-parameter family of pulse solutions are introduced in Section 2. Section 3 is devoted to a discussion of the properties of the linearized system. In particular, we are able to calculate a fundamental matrix describing the solutions of the six-dimensional first order system obtained when the spectral parameter is zero. In Section 4, we develop regular and singular expansions of the eigenfunctions in powers of the spectral parameter. In Section 5, we define the Evans function, calculate its fifth derivative at the origin, and determine its behavior for large values of the spectral parameter. The instability criterion is given at the end of Section 5. Conclusions and open questions are discussed in Section 6.
2. The coupled Klein–Gordon equations The near-threshold dynamics of an elastic rod with circular cross-section is described by the following dimensionless coupled Klein–Gordon equations [1] 2 ∂2 A ∂B 2∂ A − c = µA − A|A|2 + A , 0 2 2 ∂x ∂t ∂x
∂2 B ∂2 B ∂|A|2 − = − , ∂x ∂t 2 ∂x2
(2.1)
where A(x, t) represents the (scaled) slowly varying complex amplitude of the helical mode which grows above the bifurcation threshold, B(x, t) is the (scaled) real axial twist, c0 and µ are real constants. The pulse solutions we are interested in belong to a class of solutions of the form A = a(ξ) eiωt ,
B = b(ξ),
ξ = x − ct,
(2.2)
where a and b solve the following ordinary differential equations in ξ (c2 − c02 )a − 2icωa = a(ω2 + µ − |a|2 + b ),
(c2 − 1)b = −(|a|2 ) .
(2.3)
As discussed in [2] (see also [3]), traveling pulses form a two-parameter family of solutions to (2.1), given by ωc α2 a0 (ξ) = α sech(βξ) exp i ξ , b (ξ) = tanh (βξ), (2.4) 0 β(1 − c2 ) c2 − c02 where α2 =
2(c2 − 1) (µ(c2 − c02 ) − ω2 c02 ), c2 (c2 − c02 )
β2 =
µ(c2 − c02 ) − ω2 c02 (c2 − c02 )2
.
(2.5)
In Section 3, we will restrict ourselves to the case ω = 0, which corresponds to “non-rotating” boundary conditions. Conditions (2.5) can then be simplified into α2 =
2µ 2 (c − 1), c2
β2 =
c2
µ . − c02
The linearization of the Klein–Gordon equations (2.1) about A = 0 and B constant shows that amplitude perturbations of the form exp[i(kx + Ωt)] have a dispersion relation given by −Ω2 = µ − c02 k2 . This indicates that when µ > 0, Fourier modes with wave number k such that k2 < µ/c02 may experience growth. As a consequence, when µ > 0, the asymptotic state of the pulse solution is unstable, since the continuous spectrum of the linearization
106
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
about the pulse solution intersects the right half-plane. In what follows, we therefore only consider the case µ < 0. The following conditions should then be satisfied in order for α and β to be real c2 < c02 ,
−1 < c < 1.
(2.6)
For convenience, we will also assume that β and c0 are positive. There is no loss of generality in doing so since the pulse solution (2.4) is not affected by changing the signs of β or c0 .
3. The linear problem We start by linearizing (2.1) around the pulse solution (2.4) for ω = 0. We write the perturbed solution as A(x, t) = a0 (ξ) + u(ξ, ˜ t),
¯ A(x, t) = a¯ 0 (ξ) + v˜ (ξ, t),
B(x, t) = b0 (ξ) + w(ξ, ˜ t),
where u, ˜ v˜ and w ˜ are small. Substitution of these expressions in Eq. (2.1) and linearization give rise to a system of the form I3 O3 ∂Y , ˜ V˜ , W) ˜ T, L = = LY, Y = (u, ˜ v˜ , w, ˜ U, ˜ 2cI3 ∂ ∂t L ∂ξ ˜ is given by where O3 and I3 are the 3 × 3 zero and identity matrices, respectively and the 3 × 3 operator L 2 ∂ 2 2 ∂ −a02 (ξ) a0 (ξ) (c0 − c ) ∂ξ 2 + C0 ∂ξ 2 ∂ ∂ 2 2 2 , C0 = µ − 2|a0 (ξ)|2 + b (ξ). ˜ L= −¯a0 (ξ) (c0 − c ) 2 + C0 a¯ 0 (ξ) 0 ∂ξ ∂ξ ∂ ∂ ∂2 − a¯ 0 (ξ) + a¯ 0 (ξ) − a0 (ξ) + a0 (ξ) (1 − c2 ) 2 ∂ξ ∂ξ ∂ξ Since b0 is finite at ±∞, we allow perturbations of B which are bounded (as opposed to decaying to zero) on the real line. We therefore consider the spectral problem λY = LY , where Y is given above and (u, ˜ v˜ , w) ˜ = (u(ξ), v(ξ), w(ξ)) ∈ (H 1 × H 1 × L∞ ) ∩ (C1 × C1 × C1 ).
(3.1)
The functions u, v and w then satisfy a six-dimensional linear dynamical system of the form X = A(ξ, λ)X,
X = (u, v, w, u , v , w )T ,
where A is the 6 × 6 matrix given by 0 0 0 0 0 0 a02 (ξ) C1 − 2 2 c2 − c02 A(ξ, λ) = c − c0 2 C1 − a¯ 0 (ξ) c2 − c2 2 c − c02 0 a¯ (ξ) a (ξ) − 20 − 20 c −1 c −1
0 0 0 0
(3.2)
1 0 0 2cλ c2 − c02
0 −
λ2 c2 − 1
0 −
a¯ 0 (ξ) c2 − 1
0 1 0 0 2cλ − c02
c2 −
a0 (ξ) c2 − 1
0 0 1 a0 (ξ) c2 − c02
, a¯ 0 (ξ) c2 − c02 2cλ c2 − 1
(3.3)
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
107
with C1 = −λ2 + µ − 2|a0 (ξ)|2 + b0 (ξ) = −λ2 + C0 . We are interested in identifying bounded solutions to (3.2) for real positive values of the spectral parameter λ. When they exist, such solutions will be called eigenfunctions of the linear operator L, and the corresponding values of the spectral parameter λ will be called eigenvalues. Linear system (3.2) has non-constant coefficients and, as far as we know, cannot be solved explicitly for arbitrary values of λ. We first look at the particular case λ = 0, which can be solved completely, and second at the asymptotic system obtained by taking the limit of (3.2) as ξ → ±∞. We then discuss under what conditions the limit as ξ → ±∞ of solutions to (3.2) are solutions of this asymptotic system. 3.1. Solution to the linear problem for λ = 0 In the case ω = 0, symmetry properties of (2.3) allow us to completely solve the system X = A(ξ, 0)X,
(3.4)
where A(ξ, 0) is the matrix given by (3.3) with λ = 0. It can indeed be shown, using a well known algorithm [23], that there are only four Lie point symmetries for system (2.3). These symmetries are reductions of the symmetries of the system of PDEs (2.1) and are listed below. We use the following notation: a(ξ) and b(ξ) are solutions of ˜ system (2.3) and the new solutions a˜ (ξ) and b(ξ) are obtained by applying the symmetries to a and b. 1. Reference frame invariance: ˜ b(ξ) = b(ξ + ξ0 ),
a˜ (ξ) = a(ξ + ξ0 ),
(3.5)
where ξ0 is a real arbitrary constant. 2. Uniform shift on b: a˜ (ξ) = a(ξ),
˜ b(ξ) = b(ξ) + χ,
(3.6)
where χ is a real arbitrary constant. 3. Gauge invariance: ˜ b(ξ) = b(ξ),
a˜ (ξ) = a(ξ) eiφ ,
(3.7)
where φ is a real arbitrary constant. 4. Dilation invariance (present only when ω = 0): a˜ (ξ) = γa(γξ),
˜ b(ξ) = γb(γξ) + µ(γ 2 − 1)ξ,
(3.8)
where γ is a real constant. Each of these symmetries gives us a solution to (3.4). Indeed, consider an analytic symmetry with parameter ( ∈ R, whose action on the solution (a0 (ξ), b0 (ξ)) of (2.3) is given by a˜ 0( (ξ) = Γ1 (ξ, () = a0 (ξ) + (φ1 (ξ) + · · · ,
b˜ 0( (ξ) = Γ2 (ξ, () = b0 (ξ) + (φ2 (ξ) + · · · .
(3.9)
Since (˜a0( (ξ), b˜ 0( (ξ)) is also a solution of (2.3), the terms in (3.9) linear in ( must be a solution of the linearization of (2.3) around the solution (a0 (ξ), b0 (ξ)), i.e. (φ1 (ξ), φ2 (ξ)) must solve Eq. (3.4). Since we restricted ourselves to the case ω = 0, Eq. (2.3) then has four independent symmetries (3.5)–(3.8), which give us four linearly independent solutions to the linear system (3.4). These solutions are listed below (we
108
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
only write the first three components of the vector X; the last three components are the derivatives of the components shown): 0 X1 (ξ) = 0 , 1
a0 (ξ) X2 (ξ) = −a0 (ξ) , 0
a0 (ξ)
X3 (ξ) = a0 (ξ) , b0 (ξ)
X4 (ξ) =
a0 (ξ) + ξa0 (ξ) a0 (ξ) + ξa0 (ξ) b0 (ξ) + ξb0 (ξ) + 2µξ
. (3.10)
Note that in the limit ξ → ±∞, the solutions X2 and X3 converge to zero, X1 is constant and X4 diverges. In other words, only three of the four symmetries give solutions which are bounded at infinity. Note also that since ω = 0, a0 (ξ) in (2.4) is a real solution and thus a¯ 0 (ξ) = a0 (ξ). It turns out that the two remaining solutions in a fundamental set of solutions of (3.4) can be found by reduction of order and are given by
sinh (2βξ) ξ (ξ) + a0 4β 2 sinh (2βξ) ξ , X5 (ξ) = + −a0 (ξ) 4β 2
0
X6 (ξ) =
a0 (ξ)
sinh (2βξ) ξ + 16β 8
sinh (2βξ) ξ + 16β 8 sinh (2βξ) ξ ξµ b0 (ξ) + + 2 (1 − c2 ) 16β 8 2c a0 (ξ)
. (3.11)
These solutions diverge as ξ → ±∞. Since the linearized system at λ = 0 has only three linearly independent solutions in the class of admissible perturbations defined by (3.1), λ = 0 is an eigenvalue of L of geometric multiplicity equal to 3. In the same way as symmetries of the equations give genuine eigenfunctions, symmetries of a family of solutions can also lead to generalized eigenfunctions. More precisely, pulse solutions to (2.1), written as a first order system, ∂Y/∂t = F(Y), are of the form G(ωt)T(−ct)ψc,ω (x), where T corresponds to translational invariance (T(ct)ϕ(x) = ϕ(x + ct)), G to gauge invariance (G(ωt)ϕ(x) = exp(iωD0 t)ϕ(x), where D0 = diag(1, −1, 0, 1, −1, 0)) and ψc,ω (x) is a six component function of x parametrized by c and ω. When such a solution is substituted into (2.1), we have ∂ [G(ωt)T(−ct)ψc,ω (x)] = F(G(ωt)T(−ct)ψc,ω (x)) = G(ωt)F(T(−ct)ψc,ω (x)), ∂t i.e. iωD0 G(ωt)T(−ct)ψc,ω (x) − cG(ωt)
∂ [T(−ct)ψc,ω (x)] = G(ωt)F(T(−ct)ψc,ω (x)), ∂ξ
ξ = x − ct.
Since the action of G is multiplicative, the above equation can be re-written as iωD0 T(−ct)ψc,ω (x) − c
∂ [T(−ct)ψc,ω (x)] = F(T(−ct)ψc,ω (x)). ∂ξ
(3.12)
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
109
By taking the derivative of (3.12) with respect to c we get ∂ ∂ ∂ ∂ iωD0 T(−ct) ψc,ω (x) − [T(−ct)ψc,ω (x)] − c T(−ct) ψc,ω (x) ∂ξ ∂c ∂ξ ∂c ∂ = DF(T(−ct)ψc,ω (x)) T(−ct) ψc,ω (x) , ∂c where DF is the Jacobian of F . When we set ω = 0, the above equation implies ∂ ∂ P = − P, L ∂c ∂ξ where P is the pulse solution at ω = 0. Since the right-hand side is in the kernel of L, ∂P/∂c is a generalized eigenfunction of L belonging to the eigenvalue λ = 0. Similarly, by taking the derivative of (3.12) with respect to ω and setting ω = 0, one finds that the derivative of the pulse solution with respect to ω, taken at ω = 0, Pω is a generalized eigenfunction of L such that L(Pω ) = iD0 P. Therefore, λ = 0 is an eigenvalue of L of algebraic multiplicity 5 when ω = 0. When ω = 0, one can still solve the linear problem at λ = 0 as follows. The three symmetries (3.5)–(3.7) still give three genuine eigenfunctions. Three other linearly independent solutions can be found by variation of parameters. It turns out that the system of equations for the unknown functions of ξ decouples into a two-by-two first order system and a first order linear differential equation. The latter equation can always be solved, at least formally, and, by rescaling the independent variable and one of the dependent variables in the two-by-two system, one obtains the same two-by-two system as in the case ω = 0, which can thus also be solved. The formula for a fundamental matrix of the linear differential system in ξ is however very cumbersome and it is only when ω = 0 that the calculations presented below are tractable. The above discussion for the existence of two generalized eigenfunctions for λ = 0 also applies when ω = 0. This argument is similar to that of [22] for Hamiltonian symmetries, but is more generic in the sense that it should work for any two-parameter family of solutions of a partial differential equation for which the two parameters are due to translational and gauge invariance. 3.2. Asymptotic system We now turn to the asymptotic system obtained as ξ → ±∞. Let A∞ (λ) = lim A(ξ, λ), ξ→±∞
where A is given by (3.3). Since a0 (ξ) and b0 (ξ) both converge to zero as ξ → ±∞, we have 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 µ − λ2 2cλ 0 0 0 0 2 − c2 A∞ (λ) = c2 − c2 . c 0 0 2 2cλ µ − λ 0 0 0 0 2 2 2 2 c − c0 c − c0 2 λ 2cλ 0 0 − 2 0 0 c −1 c2 − 1
110
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
The eigenvalues νi∞ (λ) and corresponding eigenvectors v∞ i,j (λ) of this matrix can be calculated explicitly. Its eigenvalues are given by
cλ + µ(c2 − c02 ) + c02 λ2 λ λ ν1∞ (λ) = , ν2∞ (λ) = , ν3∞ (λ) = , c+1 c−1 c2 − c02
cλ − µ(c2 − c02 ) + c02 λ2 , (3.13) ν4∞ (λ) = c2 − c02 where ν3∞ (λ) and ν4∞ (λ) have geometric multiplicity 2. The corresponding eigenvectors are ∞ v1 (λ) =
0 0 1 0 0 ν1∞ (λ)
v∞ (λ) = 3,2 ∞ v4,2 (λ) =
∞ v2 (λ) =
,
(c2 − c02 )ν3∞ (λ) − 2cλ 0 0 µ − λ2 0 0 (c2 − c02 )ν4∞ (λ) − 2cλ 0 0 µ − λ2 0 0
,
0
,
0 1 0 0 ν2∞ (λ)
0
2 (c − c2 )ν∞ (λ) − 2cλ 0 3 0 ∞ , v3,1 (λ) = 0 2 µ−λ 0
0
(c2 − c2 )ν∞ (λ) − 2cλ 0 4 0 , v∞ (λ) = 4,1 0 µ − λ2 0
.
(3.14)
The convergence, as ξ → ±∞, of solutions to X = A∞ (λ)X depends on the real parts of the eigenvalues νi∞ (λ). If the real and imaginary parts of λ are denoted by λr and λi , then
√ 2cλ + R + µ(c2 − c02 ) + c02 (λ2r − λ2i ) r λ λ r r , Re(ν1∞ ) = , Re(ν2∞ ) = , Re(ν3∞ ) = √ c+1 c−1 2(c2 − c02 )
√ 2cλr − R + µ(c2 − c02 ) + c02 (λ2r − λ2i ) Re(ν4∞ ) = , √ 2(c2 − c02 ) where R=
(µ(c2 − c02 ) + c02 (λ2r − λ2i ))2 + 4c04 λ2r λ2i .
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
111
It is a simple matter of calculation to prove that the only location in the complex plane where at least one of the Re(νi∞ ) may vanish is on the imaginary axis (i.e. where λr = 0). Moreover, for λr = 0 and |λi | <
µ(c2 − c02 )/c02 ,
only two eigenvalues, ν1∞ and ν2∞ , are purely imaginary (recall that µ(c2 − c02 ) > 0 because µ < 0 and because of
(2.6)), whereas all four eigenvalues are imaginary when λr = 0 and |λi | ≥ µ(c2 − c02 )/c02 . Since the eigenvalues νi∞ are continuous functions of λ, the only place where their real part may change sign is therefore when λ crosses the imaginary axis. When λ is real and sufficiently large, the real parts of ν1∞ and ν4∞ are positive and the real parts of ν2∞ and ν3∞ are negative. Similarly, when λ is a sufficiently large real negative number, the real parts of ν2∞ and ν4∞ are positive and the real parts of ν1∞ and ν3∞ are negative. Together with the fact that Re(νi∞ ) can only change sign when λ is on the imaginary axis, these statements imply that A∞ (λ) has three eigenvalues (counting multiplicity) with strictly positive real parts and three eigenvalues with strictly negative real parts for all λ’s with non-zero real part. In the region of interest, i.e. in the right half complex plane (where λr ≥ 0), the situation can be summarized as follows: Re(ν1∞ (λ)) ≥ 0,
Re(ν2∞ (λ)) ≤ 0,
Re(ν3∞ (λ)) ≤ 0,
Re(ν4∞ (λ)) ≥ 0.
Following [8], we define the deviator to be R(ξ, λ) = A(ξ, λ) − A∞ (λ). Since the components of the matrix A(ξ, λ) converge exponentially to the components of A∞ (λ) as ξ → ±∞, it is clear that the integral ∞ |R(ξ, λ)| dx, (3.15) −∞
converges uniformly in λ on any compact subset of C. We now turn to solutions of (3.2) which are bounded either as ξ → ∞ or as ξ → −∞. It is known [24] that if the integral (3.15) converges and if the matrix A∞ (λ) is diagonalizable (although this last condition is not necessary; see [7]), solutions to (3.2) that are bounded for ξ positive behave, in the limit as ξ → ∞, like solutions of the constant coefficient system X = A∞ (λ)X. More precisely, when Re(λ) > 0 the matrix A∞ (λ) has a three-dimensional eigenspace corresponding to eigenvalues with negative real parts, and hence there is a three-dimensional space of solutions to (3.2) which are bounded as ξ → ∞. Moreover, three linearly independent solutions φi (ξ, λ), i = 1, 2, 3, to (3.2) can be chosen in this space such that ∞ (λ)ξ
lim φ1 (ξ, λ) e−ν2
ξ→∞
= v∞ 2 (λ),
∞ (λ)ξ
lim φ2 (ξ, λ) e−ν3
ξ→∞
= w2 (λ),
∞ (λ)ξ
lim φ3 (ξ, λ) e−ν3
ξ→∞
= w3 (λ), (3.16)
where w2 and w3 are any two linearly independent vectors in the eigenspace of A∞ (λ) belonging to ν3∞ (λ). A similar result holds for solutions to (3.2) which are bounded as ξ → −∞. Three such solutions φi (ξ, λ), i = 4, 5, 6, can then be chosen such that ∞ (λ)ξ
lim φ4 (ξ, λ) e−ν1
ξ→−∞
= v∞ 1 (λ),
∞ (λ)ξ
lim φ5 (ξ, λ) e−ν4
ξ→−∞
= w5 (λ),
∞ (λ)ξ
lim φ6 (ξ, λ) e−ν4
ξ→−∞
= w6 (λ), (3.17)
∞ where w5 and w6 are any two linearly independent vectors in the eigenspace generated by v∞ 4,1 (λ) and v4,2 (λ). Regarding the analyticity of the φi ’s, one must first observe that there exists a domain containing the positive real
112
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
axis λr > 0, in which the eigenvectors (3.14) and eigenvalues (3.13) are analytic and in which the number of eigenvalues with distinct real parts does not change. Because of this and because the convergence of the integral (3.15) is uniform in λ on any compact subset of C, the φi are analytic in λ ∈ R, λ > 0, for each fixed ξ [24]. When λ = 0, the eigenvalues ν1∞ (λ) (ν1∞ (λ) ≥ 0 for λ ∈ R+ ) and ν2∞ (λ) (ν2∞ (λ) ≤ 0 for λ ∈ R+ ) vanish. The above argument can be extended to show that there exist solutions φi whose asymptotic behavior is given by (3.16) and (3.17) and which are analytic in λ in a neighborhood of λ = 0 if R(ξ, λ) = O(exp(−γ|ξ|))
as |ξ| → ∞, γ > 0,
and if γ > |Re(ν1∞ (λ) − ν2∞ (λ))|. Such solutions are constructed in the next section. This illustrates how the Evans function can be extended in an analytic fashion to values of the spectral parameter for which the asymptotic matrix has eigenvalues with zero real part (in the L2 case, this occurs when the spectral parameter enters the continuous spectrum). Analytic continuation of the Evans function across the continuous spectrum has been the topic of several recent papers (see for instance [11,12,19,25]) and is addressed by the Gap Lemma [11,12]. The above argument, together with the calculations of Section 4, provides an alternative proof of the Gap Lemma in the case where A∞ (λ) is diagonalizable for Re(λ) = 0. We also note that since the eigenvalues of A∞ (λ) are analytic in λ in a neighborhood of λ = 0, the Evans function defined in Section 5.1 does not have a branch point at the origin. The reader is referred to [19] for a situation where an eigenvalue lies on a branch point of the Evans function. Finally, since the matrix A in (3.3) is real when λ is real, the φi can also be chosen to be real in this particular case.
4. Asymptotic expansions of solutions for small values of the spectral parameter This section is devoted to the behavior, for small real λ, of the solutions φi (ξ, λ) defined above. We know a fundamental matrix for the system X = A(ξ, 0)X, obtained when λ = 0, and we look for asymptotic expansions of the φi ’s as λ → 0, which are uniformly valid in ξ. Regular expansions of the form (0)
(1)
(2)
φi (ξ, λ) = φi (ξ) + φi (ξ)λ + φi (ξ)λ2 + · · · ,
(4.1)
are sought for φ2 , φ3 (for ξ ≥ 0), φ5 and φ6 (for ξ ≤ 0) since the corresponding eigenvalues νi∞ (λ) have non-zero real parts when λ = 0, whereas singular expansions are sought for φ1 and φ4 . The latter are of the form ∞ (λ)ξ
φ1 (ξ, λ) = eν2
∞ (λ)ξ
φ4 (ξ, λ) = eν1
(0) (1) (2) (φ˜ 1 (ξ) + φ˜ 1 (ξ)λ + φ˜ 1 (ξ)λ2 + · · · ), (0)
(1)
(2)
(φ˜ 4 (ξ) + φ˜ 4 (ξ)λ + φ˜ 4 (ξ)λ2 + · · · ),
(4.2)
(j) (j) (j) where the φ˜ i are such that limξ→∞ φ˜ 1 (ξ) and limξ→−∞ φ˜ 4 (ξ) exist and are finite (see also [12,19] for singular (0) (0) expansions near a branch point of the Evans function). The φi and φ˜ i satisfy the linear homogeneous system (j) (j) (3.4). The higher order terms φi (resp. φ˜ i ) with j > 0 solve a linear system of the form X − A(ξ, 0)X = f(ξ), (k) (k) where the inhomogeneous part is determined by terms found at preceding orders, i.e. by the φi (resp. φ˜ i ) with k < j. Since the homogeneous system can be solved completely and has solutions given by (3.10) and (3.11), the inhomogeneous systems obtained at each order can be solved by the method of variation of parameters. (0) (0) At order zero, the φi and φ˜ i are linear combinations of the Xi ’s. One can check that ∞ lim X1 (ξ) = v∞ 1 (0) = v2 (0) = e3 ,
ξ→±∞
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
113
where e3 = (0, 0, 1, 0, 0, 0)T and that X2 and X3 satisfy 2α ∞ (v∞ 3,1 (0) − v3,2 (0)), ξ→∞ − c02 ) 2α ∞ ∞ (v∞ lim X3 (ξ) e−ν3 (0)ξ = 3,1 (0) + v3,2 (0)), ξ→∞ c2 − c02 2α ∞ ∞ lim X2 (ξ) e−ν4 (0)ξ = (−v∞ 4,1 (0) + v4,2 (0)), 2 ξ→−∞ β(c − c02 ) 2α ∞ ∞ lim X3 (ξ) e−ν4 (0)ξ = (v∞ 4,1 (0) + v4,2 (0)). 2 ξ→−∞ c − c02 ∞ (0)ξ
lim X2 (ξ) e−ν3
=
β(c2
It is thus possible to choose the first terms in the expansions (4.1) and (4.2) to be (0) (0) φ˜ 1 (ξ) = φ˜ 4 (ξ) = X1 (ξ),
(0)
(0)
(0)
φ2 (ξ) = φ5 (ξ) = X2 (ξ),
(0)
φ3 (ξ) = φ6 (ξ) = X3 (ξ),
(4.3)
with the wi in (3.16) and (3.17) then given by 2α ∞ (v∞ 3,1 (λ) − v3,2 (λ)), β(c2 − c02 ) 2α ∞ w5 (λ) = (−v∞ 4,1 (λ) + v4,2 (λ)), 2 β(c − c02 )
w2 (λ) =
2α ∞ (v∞ 3,1 (λ) + v3,2 (λ)), c2 − c02 2α ∞ w6 (λ) = (v∞ 4,1 (λ) + v4,2 (λ)). 2 c − c02
w3 (λ) =
(4.4)
(1)
We now look at first order corrections. As mentioned before, the φi satisfy inhomogeneous equations that can be (0) (1) solved once the φi are chosen. In the case of φ2 , the most general solution of the corresponding inhomogeneous linear equation which decays to zero as ξ → ∞ is given by (1)
(1)
φ2 (ξ) = φ2,part (ξ) + c2,2 X2 (ξ) + c2,3 X3 (ξ),
(4.5) (1)
where c2,2 and c2,3 are arbitrary real constants and φ2,part (ξ) is a particular solution found using variation of constants αc(2βξ + 1) sech(βξ) 2β(c2 − c02 ) αc(2βξ + 1) sech(βξ) (1) φ2,part (ξ) = − . 2β(c2 − c02 ) 0
(4.6)
As before, we only show the first three components of each vector. The constants c2,2 and c2,3 are determined by condition (3.16) with w2 given in (4.4). The behavior near the origin of the Evans function defined in the next section (1) will depend on φ2,part (ξ) but will not be affected by the particular values of c2,2 and c2,3 . We will be only interested in finding the sign of the lowest non-zero derivative of the determinant (5.1) at λ = 0; adding linear combinations (j) (0) of the φi to the φi with j > 0 and i = 2, 3, 5, 6 only alters higher order derivatives. We therefore do not evaluate c2,2 and c2,3 . (1) Similarly, φ3 can be determined to be (1)
(1)
φ3 (ξ) = φ3,part (ξ) + c3,2 X2 (ξ) + c3,3 X3 (ξ),
(4.7)
114
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
where
α(c2 (1 − c2 )(1 + 2βξ) sinh (βξ) + 2(c02 − c2 ) e−βξ )
2c cosh 2 (βξ)(c2 − 1)(c2 − c02 ) α(c2 (1 − c2 )(1 + 2βξ) sinh (βξ) + 2(c02 − c2 ) e−βξ ) (1) φ3,part (ξ) = 2c cosh 2 (βξ)(c2 − 1)(c2 − c02 ) µ((c2 − 1)(c2 − 2c02 ) e−2βξ + 2c2 (1 − c02 + βξ(1 − c2 ))) βc3 cosh 2 (βξ)(c2 − 1)(c2 − c02 )
.
(4.8)
The first order terms in φ5 (ξ, λ) and φ6 (ξ, λ) must converge to zero as ξ → −∞, i.e. (j)
lim φi (ξ) = 0,
ξ→−∞
i = 5, 6,
(4.9)
and are determined in a similar fashion. We obtain (1)
(1)
(1)
(1)
φ5 (ξ) = φ2,part (ξ) + c5,2 X2 (ξ) + c5,3 X3 (ξ),
(4.10)
and φ6 (ξ) = φ3,part (ξ) − 4
µ(c2 − 2c02 ) βc3 (c2 − c02 )
X1 (ξ) + c6,2 X2 (ξ) + c6,3 X3 (ξ),
(4.11) (1)
where the term in X1 was added to make sure that condition (4.9) is satisfied by φ6 . Recall that X1 , given in (3.10) (1) (1) is simply the constant vector e3 . Finally, the functions φ˜ 1 (ξ) and φ˜ 4 (ξ) are found to be (1) φ˜ 1 (ξ) = −
ξ X4 (ξ) e3 + + c1,1 X1 (ξ) + c1,2 X2 (ξ) + c1,3 X3 (ξ), c−1 2µ(c − 1)
where X4 is given in (3.10) and ξ X4 (ξ) (1) φ˜ 4 (ξ) = − e3 + + c4,1 X1 (ξ) + c4,2 X2 (ξ) + c4,3 X3 (ξ). c+1 2µ(c + 1) (i)
(i)
In order to have more uniform notation, we also give expressions for φ1 (ξ) and φ4 (ξ) obtained from (4.2) by expanding the exponential in powers of λ. We have (0) (0) φ1 (ξ) = φ˜ 1 (ξ),
(0) (0) φ4 (ξ) = φ˜ 4 (ξ), ξ ˜ (0) X4 (ξ) (1) (1) φ1 (ξ) = φ˜ 1 (ξ) + φ1 (ξ) = + c1,1 X1 (ξ) + c1,2 X2 (ξ) + c1,3 X3 (ξ), c−1 2µ(c − 1) X4 (ξ) (1) φ4 (ξ) = + c4,1 X1 (ξ) + c4,2 X2 (ξ) + c4,3 X3 (ξ). 2µ(c + 1)
(4.12)
Terms at order two are given in Appendix A. We will not need to go any further in the asymptotic expansions to be
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
115
able to characterize the behavior of the Evans function near the origin. To summarize, we have X4 (ξ) λ+ c1,i Xi (ξ)λ + O(λ2 ), 2µ(c − 1) 3
φ1 (ξ, λ) = X1 (ξ) +
i=1
(1)
φ2 (ξ, λ) = X2 (ξ) + φ2,part (ξ)λ +
3
(2)
c2,i Xi (ξ)λ + φ2,part (ξ)λ2 +
i=2
+
3
3 i=2
(1)
c2,i φi,part (ξ)λ2
e2,i Xi (ξ)λ2 + O(λ3 ),
i=2 (1)
φ3 (ξ, λ) = X3 (ξ) + φ3,part (ξ)λ +
3
(2)
c3,i Xi (ξ)λ + φ3,part (ξ)λ2 +
i=2
+
3
3 i=2
(1)
c3,i φi,part (ξ)λ2
e3,i Xi (ξ)λ2 + O(λ3 ),
i=2
X4 (ξ) c4,i Xi (ξ)λ + O(λ2 ), λ+ 2µ(c + 1) 3
φ4 (ξ, λ) = X1 (ξ) +
i=1
(1)
φ5 (ξ, λ) = X2 (ξ) + φ2,part (ξ)λ +
3
(2)
c5,i Xi (ξ)λ + φ5,part (ξ)λ2
i=2
+
3 i=2
(1)
c5,i φi,part (ξ)λ2 +
φ6 (ξ, λ) = X3 (ξ) + +
3 i=2
(1)
(1) φ3,part (ξ) − 4
(1)
3
e5,i Xi (ξ)λ2 + O(λ3 ),
i=2
µ(c2 − 2c02 ) βc3 (c2
c6,i φi,part (ξ)λ2 +
3
− c02 )
X1 (ξ) λ +
3
(2)
c6,i Xi (ξ)λ + φ6,part (ξ)λ2
i=2
e6,i Xi (ξ)λ2 + O(λ3 ),
(4.13)
i=2 (2)
where the φi,part are given in (4.6) and (4.8) and the φi,part are given in Appendix A. Terms of order two or higher in λ that will not be needed in the next section have been omitted. The sums reflect the fact that, at each order greater than zero, convenient linear combinations of X1 , X2 and X3 should be included in order to ensure that conditions (3.16) and (3.17) are satisfied. However, as mentioned above, the actual values of the ci,j and ei,j are irrelevant since, as shown in Section 5.2, they do not change the value of the first non-zero derivative of the Evans function (5.1) at the origin.
5. Evans function 5.1. Definition We defined eigenvalues as values of λ for which the system (3.2) has a solution which is bounded as ξ → ±∞, as defined by (3.1). Equivalently, λ0 is an eigenvalue if and only if the space of solutions bounded as ξ → +∞, spanned by {φ1 , φ2 , φ3 }, and the space of solutions bounded as ξ → −∞, spanned by {φ4 , φ5 , φ6 }, have an intersection of
116
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
strictly positive dimension when λ = λ0 . This indeed implies that there is a non-trivial solution to (3.2) with λ = λ0 , which is bounded as ξ → ±∞. The most straightforward way to test whether these two spaces of solutions intersect non-trivially is to calculate the determinant of the two spanning sets. Since it suffices to calculate this determinant at ξ = 0, we define ui (λ) = φi (0, λ),
i = 1, 2, . . . , 6.
Then λ0 is an eigenvalue of (3.2) if and only if the function D(λ) = det(u1 (λ), u2 (λ), u3 (λ), u4 (λ), u5 (λ), u6 (λ)),
(5.1)
evaluated at λ = λ0 is zero. The function D(λ) is called the Evans function [4–7,11–13,19,25]. It is analytic in a neighborhood of the positive real axis and it is real for λ real. In what follows, we study the behavior of the Evans function at the origin and for large values of the spectral parameter λ. 5.2. Behavior of the Evans function near the origin In Section 4, we considered expansions of the φi ’s near λ = 0. The functions ui (λ) can also be expanded as (0)
(1)
(2)
ui (λ) = ui + ui λ + ui λ2 + · · · , (j)
(j)
where ui = φi (0). From (4.13), one can see that u5 (λ) − u2 (λ) = λ
3 i=2
(0)
d2,i ui + O(λ2 ),
u6 (λ) − u3 (λ) = λ
3 i=1
(0)
d3,i ui + O(λ2 ),
where d2,i = c5,i − c2,i , d3,i = c6,i − c3,i for i = 2, 3, and d3,1 = −4(µ(c2 − 2c02 ))/(βc3 (c2 − c02 )). One can therefore re-write D(λ) as 3 3 D(λ) = det u1 (λ), u2 (λ), u3 (λ), u4 (λ), u5 (λ) − λ d2,i ui (λ), u6 (λ) − λ d3,i ui (λ) i=2
i=1
≡ det(u1 (λ), u2 (λ), u3 (λ), u4 (λ), u∗5 (λ), u∗6 (λ)),
(5.2)
where u∗5 (λ) − u2 (λ) = O(λ2 ),
u∗6 (λ) − u3 (λ) = O(λ2 ).
(5.3)
The Evans function (5.1) then becomes D(λ) = D(0) + D(1) λ + D(2) λ2 + · · · ,
(5.4) (j)
where each D(n) can be calculated in terms of the ui (i ) (i ) (i ) (i ) ∗(i ) ∗(i ) D(n) = det(u1 1 , u2 2 , u3 3 , u4 4 , u5 5 , u6 6 ),
(5.5)
|i|=n
with i = (i1 , i2 , i3 , i4 , i5 , i6 ), |i| = i1 + i2 + i3 + i4 + i5 + i6 . Because of the equalities given in (4.3), the two (0) (0) (0) (0) ∗(0) ∗(0) spaces spanned by {u1 , u2 , u3 } and by {u4 , u5 , u6 } are identical since (0)
(0)
u1 = u4 ,
(0)
(0)
∗(0)
u2 = u5 = u5
,
(0)
(0)
∗(0)
u3 = u6 = u6
.
(5.6)
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
117
It is thus easy to conclude from (5.6) that at least three of the ij ’s must be non-zero in order for one of the terms in the sum (5.5) to be non-zero. Hence D(0) = D(1) = D(2) = 0. This fact is not surprising [7,13] since we indicated before that the geometric multiplicity of the eigenvalue λ = 0 is equal to 3. From (5.6), it is also easy to combine the non-zero terms in the expression for D(3) given by (5.5) to obtain (1)
(1)
(1)
∗(1)
(1)
∗(1)
D(3) = det(u1 − u4 , u2 − u5 (1)
∗(1)
(1)
∗(1)
, u3 − u6
(0)
(0)
(0)
, u1 , u2 , u3 ).
From (5.3), u2 = u5 and u3 = u6 , and D(3) is therefore equal to zero. As for the fourth term in the expansion (5.4), non-zero contributions to D(4) containing only terms of order zero and one add up to (1)
(0)
(0)
(1)
∗(1)
− u2 , u6
(1)
∗(1)
− u3 ) + det(u1 , u2 , u3 , u4 − u1 , u5
(0)
(0)
(1)
(1)
(1)
∗(1)
− u2 , u6
det(u1 , u2 , u3 , u4 , u5
+ det(u1 , u2 , u3 , u4 − u1 , u5
(1)
(1)
(0)
∗(1)
(1)
(0)
(1)
(1)
(2)
(2)
(2)
(1)
∗(1)
(1)
∗(1)
, u3 − u6
(0)
(1)
(2)
∗(2)
, u3 − u6
(1)
(1)
(1)
∗(1)
, u3 − u6
+ det(u1 − u4 , u2 − u5
(1)
− u3 )
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
is given by
, u1 , u2 , u3 )
(1)
+ det(u1 − u4 , u2 − u5
∗(1)
, u6
),
which is zero. The contribution to D(4) coming from non-zero terms containing any ui det(u1 − u4 , u2 − u5
∗(1)
(1)
∗(1)
, u1 , u2 , u3 )
(2)
∗(2)
, u1 , u2 , u3 ),
which is also zero because of relation (5.3). Even though λ = 0 is not an isolated eigenvalue of L, the vanishing of D(3) and D(4) confirms that the algebraic multiplicity of λ = 0 is equal to 5 [7]. This can indeed be understood as follows. One can easily check that an equation of the form φ(0) = A(ξ, 0)φ(0) is equivalent to LY0 = 0, where Y0 = (φ(0) [1], φ(0) [2], φ(0) [3], 0, 0, 0)T and φ(0) [i] refers to the ith component of φ(0) (ξ). Similarly, the equation ∂A(ξ, λ) (1) (1) = A(ξ, 0)φ + φ(0) φ ∂λ λ=0 reads LY1 = Y0 with Y1 = (φ(1) [1], φ(1) [2], φ(1) [3], φ(0) [1], φ(0) [2], φ(0) [3])T and ∂A(ξ, λ) 1 ∂2 A(ξ, λ) (2) (2) (1) = A(ξ, 0)φ + φ + φ(0) φ ∂λ λ=0 2 ∂λ2 λ=0 reads LY2 = Y1 with Y2 = (φ(2) [1], φ(2) [2], φ(2) [3], φ(1) [1], φ(1) [2], φ(1) [3])T . As a consequence, for each bounded φ(0) , the existence of a solution φ(1) bounded at +∞ and −∞ is equivalent to increasing by 1 the algebraic multiplicity of the eigenvalue λ = 0. If both φ(1) and φ(2) are bounded, the algebraic multiplicity of λ = 0 is (1) (1) (0) increased by 2. Here, φ2 , which converges at +∞, and φ5 , which converges at −∞, are such that φ2 (ξ) = (0) (1) (1) φ5 (ξ) = X2 (ξ). The first relation in (5.3) indicates that φ2 and φ5 , corrected by linear combinations of the Xi ’s, i = 1, 2, 3, match at ξ = 0, which implies that there exists a bounded φ(1) for φ(0) = X2 . Similarly, the second relation in (5.3) implies that there is a bounded φ(1) for φ(0) = X3 . A consequence of (5.3) is therefore that the algebraic multiplicity of λ = 0 is equal to 5. In terms of the derivatives of D(λ) at λ = 0, Eq. (5.3) imply that D(3) and D(4) both vanish and that λ = 0 is a zero of the Evans function of order 5. As in the standard case [7], the order of the zero of the Evans function at λ = 0 is therefore equal to the algebraic multiplicity of this eigenvalue. In the
118
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
discussion of Section 3.1, we identified the two generalized eigenfunctions as the derivatives of the pulse solution with respect to ω and to c, respectively, taken at ω = 0. The first non-zero term in the expansion (5.4) is the sixth one. One can check that the terms in D(5) containing (2) at most one ui add up to zero and that D(5) is given by (1)
(1)
(2)
∗(2)
D(5) = det(u1 − u4 , u2 − u5
(2)
∗(2)
, u3 − u6
(0)
(0)
(0)
, u1 , u2 , u3 ).
Now (2)
∗(2)
u2 − u5
(2)
(2)
= u2 − u5 +
3 i=2
−
3 i=2
(1)
d2,i ui
(1) c5,i φi,part (0) −
(2)
= φ2,part (0) +
i=2
3 i=2
(0) e5,i ui
(1)
+ (c5,3 − c2,3 ) φ3,part (0) + (2)
(2)
3 i=2
(1)
c2,i φi,part (0) +
+ (c5,2 − c2,2 )
3 i=2
= φ2,part (0) − φ5,part (0) +
3
3 i=2
(1) φ2,part (0) +
3 i=2
(0)
(2)
e2,i ui − φ5,part (0) (0) c2,i ui
(0)
c3,i ui
(0)
(e2,i − e5,i + (c5,2 − c2,2 )c2,i + (c5,3 − c2,3 )c3,i )ui ,
and, similarly, (2)
∗(2)
u3 − u6
(2)
(2)
= u3 − u6 +
3 i=1
(2)
(1)
d3,i ui
(2)
= φ3,part (0) − φ6,part (0) + µ(c2
− 2c02 ) −4 βc3 (c2 − c02 )
3 i=2
(0)
(e3,i − e6,i + (c6,2 − c3,2 )c2,i + (c6,3 − c3,3 )c3,i )ui
3 X4 (0) (0) . + c1,i ui 2µ(c − 1) i=1
As a consequence (1)
(1)
(2)
(2)
(2)
(2)
(0)
(0)
(0)
D(5) = det(u1 − u4 , φ2,part (0) − φ5,part (0), φ3,part (0) − φ6,part (0), u1 , u2 , u3 ) µ(c2 − 2c02 ) X4 (0) (1) (1) (2) (2) (0) (0) (0) + det u1 − u4 , φ2,part (0) − φ5,part (0), −4 , u , u2 , u3 . βc3 (c2 − c02 ) 2µ(c − 1) 1 Since (1)
(1)
u1 − u4 =
X4 (0) X4 (0) (0) − + (c1,i − c4,i )ui , 2µ(c − 1) 2µ(c + 1) 3
i=1
the second determinant in this expression is zero and an evaluation of the first determinant gives D(5) =
128 c02 µ2 (3c02 (3c2 − 2c02 ) − c4 (c02 + 2)) . 3 c6 (c2 − c02 )4
(5.7)
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
119
5.3. Behavior of the Evans function for large λ In order to study the behavior of the Evans function D(λ) as λ → ∞, we follow [13] and introduce the change of variable ζ = λξ,
(5.8)
which does not change the sign of the Evans function for λ ∈ R, λ > 0. Then Eq. (3.2) becomes ˜ λ)X, X = A(ζ,
X = (u, v, w, u , v , w )T ,
where the prime now means derivative with respect to ζ and 0 0 0 0 0 0 0 0 0 2 a (ζ/λ) C1 − 0 0 2 2 2 2 ˜ λ) = λ (c − c0 ) λ (c2 − c02 ) A(ζ, a2 (ζ/λ) C1 − 0 0 λ2 (c2 − c2 ) 2 (c2 − c2 ) λ 0 0 a (ζ/λ) a0 (ζ/λ) 1 − 20 2 − 2 2 2 λ (c − 1) λ (c − 1) c −1 with
(5.9)
c2
1 0 0
0 1 0
0 0 1
2c − c02
0
a0 (ζ/λ) λ(c2 − c02 )
2c c2 − c02
a0 (ζ/λ) λ(c2 − c02 )
0 −
a0 (ζ/λ) λ(c2 − 1)
−
a0 (ζ/λ) λ(c2 − 1)
2c −1
,
c2
2 ζ ζ C1 = −λ + µ − 2 a0 + b0 . λ λ 2
Under the change of variable (5.8), the solutions φi to (3.2) defined in Section 3 give rise to solutions of system (5.9) ζ φ˜ i (ζ, λ) = φi ,λ , λ ˜ λ) and the Evans function D(λ) can be expressed in terms of the φ˜ i (0, λ). As λ → +∞, λ ∈ R+ , the matrix A(ζ, ˜ 0 . Because the system (5.9) has an exponential dichotomy for λ converges uniformly toward a constant matrix A ˜ λ)X are uniformly close to the solutions of the system X = A ˜ 0 X [13]. The sign of large, solutions of X = A(ζ, D(λ) for large positive λ can therefore be found by calculating the Evans function for this system, provided the ˜0 basis vectors are arranged in the same order. This can easily be accomplished by comparing the eigenvectors of A with the images, under the change of variable (5.8), of the eigenvectors defined in (3.14). More precisely, we have ∞ lim φ˜ 1 (ζ, λ) e−˜ν2 (λ)ζ = v˜ ∞ 2 (λ),
ζ→∞
∞ lim φ˜ 2 (ζ, λ) e−˜ν3 (λ)ζ = w ˜ 2 (λ),
ζ→∞
∞ lim φ˜ 3 (ζ, λ) e−˜ν3 (λ)ζ = w ˜ 3 (λ),
ζ→∞
(5.10) and ∞ lim φ˜ 4 (ζ, λ) e−˜ν1 (λ)ζ = v˜ ∞ 1 (λ),
ζ→−∞
∞ lim φ˜ 5 (ζ, λ) e−˜ν4 (λ)ζ = w ˜ 5 (λ),
ζ→−∞
∞ lim φ˜ 6 (ζ, λ) e−˜ν4 (λ)ζ = w ˜ 6 (λ),
ζ→−∞
(5.11)
120
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
where the eigenvalues ν˜ i∞ and the eigenvectors v˜ ∞ i are related to the eigenvalues (3.13) and eigenvectors (3.14) of A∞ (λ) by ν˜ i∞ (λ) =
νi∞ (λ) , λ
∞ (˜v∞ i (λ))j = (vi (λ))j ,
On the other hand, the matrix 0 0 0 0 0 0 1 0 ˜ 0 = − c2 − c2 A 0 1 0 − 2 c − c02 0 0
0 0 0
1 0 0 2c c2 − c02
0 0 −
c2
(˜v∞ i (λ))j =
j = 1, 2, 3,
0
1 −1
c2
0
0 1 0
0 0 1
0
0
2c − c02
0
0
2c −1
(v∞ i (λ))j , λ
j = 4, 5, 6.
c2
has eigenvalues ν˜ i0 ν˜ 10 =
1 , c+1
ν˜ 20 =
1 , c−1
ν˜ 30 =
1 , c − c0
ν˜ 40 =
1 , c + c0
and eigenvectors 0 0 1 0 v˜ 1 = 0 , 0 ν˜ 10
v˜ 04,1
0 0 1 0
0
2 (c − c02 )˜ν30 − 2c 0 0 0 v˜ 2 = v˜ 3,1 = , , 0 0 −1 ν˜ 20 0 0 (c2 − c02 )˜ν40 − 2c (c2 − c2 )˜ν0 − 2c 0 4 0 0 0 , 0 = v˜ 4,2 = , −1 0 0 −1 0 0
v˜ 03,2
=
(c2 − c02 )˜ν30 − 2c
where, as before, ν˜ 30 and ν˜ 40 have multiplicity two. It can easily be checked that, as λ → ∞ v˜ ∞ ˜ 0i , i (λ) ∝ v
i = 1, 2,
and
w ˜ i (λ) ∝ w ˜ 0i ,
i = 2, 3, 5, 6,
where the ∝ symbol indicates that the two vectors point in the same direction, and 2α (˜v0 − v˜ 03,2 ), β(c2 − c02 ) 3,1 2α w ˜ 05 = (−˜v04,1 + v˜ 04,2 ), 2 β(c − c02 ) w ˜ 02 =
2α (˜v0 + v˜ 03,2 ), c2 − c02 3,1 2α w ˜ 06 = (˜v0 + v˜ 04,2 ). 2 c − c02 4,1
w ˜ 03 =
0 0 −1 0 0
,
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
121
Therefore, as λ → ∞, the sign of the Evans function D(λ) is the same as the sign of ˜ = det[˜v02 , w D ˜ 02 , w ˜ 03 , v˜ 01 , w ˜ 05 , w ˜ 06 ] =
512c02 α4 µ(c2 − 1)(c2 − c02 )3
,
(5.12)
which is negative. 5.4. Instability criterion We have just shown that the Evans function D(λ) is negative for λ real and sufficiently large. Therefore, if the expression given in (5.7) happens to be positive, the Evans function must have at least one zero on the positive real line. The pulse solution (2.4) is thus unstable if T ≡ 3c02 (3c2 − 2c02 ) − c4 (c02 + 2) > 0. It is only a matter of calculations to prove that, together with conditions (2.6), T > 0 can be re-written as
2 9 − 33 − 24c02 c0 K≡ < c2 < c02 < 1. 2 c02 + 2
(5.13)
This defines a non-empty set of pulse velocities c since K < c02 when c02 < 1. In the case of Fig. 1, c 0.6124, c02 = 12/5 and T = −27.078 < 0, so our criterion does not indicate instability. In fact, the numerical calculation shows that in this case the pulse is stable. At the bifurcation, when T = 0, the algebraic multiplicity of λ = 0 is (2) − φ(2) increased by 1 and is then equal to 6. A look at the formulae given in the appendix show that φ6,part 3,part is a (2) linear combination of X1 , X2 , X3 and X4 when T = 0 (since the coefficient of X6 in φ6,part vanishes). Since the (2) − φ(2) difference φ4(1) − φ1(1) also involves X1 , X2 , X3 and X4 , one can find a linear combination of φ6,part 3,part and (1) (1) φ4 − φ1 which only depends on X1 , X2 and X3 , i.e. which is bounded at ±∞. In agreement with the discussion of Section 5.2, this implies that there exists a bounded Y such that L3 Y = 0.
6. Conclusions In this article, we have considered a one-parameter family of pulse solutions (2.4) to two coupled non-linear Klein–Gordon equations (2.1). Using symmetry properties of (2.1), we were able to find an expansion of the Evans function associated with the linearization of (2.1) about these pulse solutions, for small values of the spectral parameter. The first non-zero derivative of the Evans function at the origin was obtained in (5.7) and the sign of the Evans function for λ real sufficiently large was shown to be negative in (5.12). A parity argument was then used to show instability of the pulse when criterion (5.13) holds. The calculations performed in this paper were not algorithmic and several difficulties had to be overcome. First, we had to deal with a linear system (3.2) for which the spaces of solutions bounded at +∞ and −∞ were both of dimension 3 (see however [11,14,15,17,21], for examples with stable spaces of dimension 2). As far as we know, such a situation was never considered in the study of a specific example. Moreover, because the linearized system does not have an exponential dichotomy when λ = 0, singular expansions of the solutions in powers of the spectral parameter had to be sought. As mentioned above, such expansions are typically used near a branch point of the Evans function [12,19]. The loss of exponential dichotomy at λ = 0 corresponds to the vanishing of the spectral gap [11,12] of the stable and unstable subspaces of the asymptotic matrix and is for instance a standard phenomenon in the case of conservation laws [11]. Finally, characterizing the behavior of the Evans function near the origin was
122
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
not a simple task since the various symmetries of the coupled non-linear Klein–Gordon equations and of its family of pulse solutions implied that the Evans function, as well as its first four derivatives, vanish at the origin. Criterion (5.13) guarantees instability but it is very likely that some pulse solutions of (2.1) are also unstable when this condition is not satisfied. In particular, numerical simulations have shown the presence of a marginal instability related to the dilation invariance (see Eq. (3.8)), in which B grows linearly in space. Since perturbations associated with this instability are clearly not bounded on the real line, they are not considered in the present study. As for other interesting problems, we would like to point out that a set of equations generalizing (2.1) for rods with non-circular cross-section was derived in [26]. The method discussed here could also be extended to pulse solutions of these generalized equations.
Acknowledgements We are grateful to M. Agrotis, N. Ercolani, A. Espiñola-Rocha, A. Goriely and R. Indik for useful discussions. Most of the calculations presented in this paper would not have been possible without the help of MAPLE (Waterloo Maple Inc.). One of the authors, SL acknowledges a postdoctoral fellowship from NSERC (National Science and Engineering Research Council) of Canada. This material is based upon work supported by the National Science Foundation under Grant No. DMS-0075827 to JL.
Appendix A Using the method of variation of parameters, we first find the following particular solutions to the inhomogeneous (2) (2) equations satisfied by φ2 and φ3 (again, we only show the first three components of each vector):
α(c02 e2βξ + 2β2 c2 ξ 2 + 2ξβ(c2 + c02 ) + c2 − c02 )
4µ(c2 − c02 ) cosh (βξ) (2) φˆ 2,part (ξ) = α(c02 e2βξ + 2β2 c2 ξ 2 + 2ξβ(c2 + c02 ) + c2 − c02 ) − 4µ(c2 − c02 ) cosh (βξ)
,
0 α(γ1 e−βξ + γ2 eβξ + γ3 e−3βξ ) 48βc2 (c2 − c02 )3 (c2 − 1)2 cosh 2 (βξ)
α(γ1 e−βξ + γ2 eβξ + γ3 e−3βξ ) ˆφ(2) (ξ) = 3,part 48βc2 (c2 − c02 )3 (c2 − 1)2 cosh 2 (βξ) δ1 cosh 2 (βξ) ln(1 + e−2βξ ) + δ2 e−2βξ + δ3 + δ4 e2βξ 6c4 (c2 − c02 )2 (c2 − 1)2 cosh 2 (βξ)
,
where γ1 = 3c2 (c2 − 1)(4c2 µ(c2 − 1)ξ 2 + 4β(c2 − 5)(c2 − c02 )2 ξ − (c2 − c02 )(5c4 − 7c2 + 4c2 c02 − 2c02 )), γ2 = −12c4 µ(c2 − 1)2 ξ 2 − 12c2 β(c2 − 1)2 (c2 − c02 )2 ξ +3(c2 − c02 )(5c8 − 16c6 − 2c6 c02 − 2c4 c02 + 27c4 + 12c2 c04 − 28c2 c02 + 4c04 ),
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
123
γ3 = 2(c2 − 1)(c2 − c02 )(c4 c02 + 2c4 − 9c2 c02 + 6c04 ), δ1 = 4(c2 − 1)(c2 − c02 )(6c04 − c4 + 7c4 c02 − 3c2 c02 − 9c2 c04 ), δ2 = 3c2 (c2 − 1)2 (c2 − c02 )(2βξ(c2 − 2c02 ) − c02 ), δ3 = −6µc4 (c2 − 1)2 ξ 2 − 6βc2 (c2 − 1)(c2 − c02 )(c2 c02 − 2c2 + c02 )ξ + 21 (c2 − c02 )(9c8 − 16c6 + 10c6 c02 − 70c4 c02 + 31c4 + 12c2 c02 + 48c2 c04 − 24c04 ), δ4 = 3(c2 − 1)(c2 − c02 )(2c02 − c2 )(c4 − 3c2 + 2c02 ). Particular solutions which converge as ξ → ∞ are given by (2)
(2)
φ2,part (ξ)=φˆ 2,part (ξ) − 2
c02 β (c2 − c02 )2
X5 (ξ),
(2)
(2)
φ3,part (ξ) = φˆ 3,part (ξ)−2 (2)
(2c02 − c2 )(c4 − 3c2 + 2c02 ) c4 (c2 − c02 )(c2 − 1)
X1 (ξ).
(2)
Particular solutions to the inhomogeneous equations satisfied by φ5 and φ6 , which converge as ξ → −∞, are given by (2) (2) φ5,part (ξ) = φˆ 2,part (ξ), (2)
(2)
φ6,part (ξ) = φˆ 3,part (ξ) + +
c02 (2µc2 − α2 c2 − 2µ) µc2 (c2 − 1)(c2 − c02 )
X1 (ξ) +
2 c4 c02 − c4 + 3c2 c02 − 3c04 X4 (ξ) 3 βc2 (c2 − 1)(c2 − c02 )2
8 c4 c02 + 2c4 − 9c2 c02 + 6c04 X6 (ξ). 3 βc2 (c2 − 1)(c2 − c02 )2
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
A. Goriely, M. Tabor, Nonlinear dynamics of filaments. II. Nonlinear analysis, Physica D 105 (1997) 45–61. J. Lega, A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132 (1999) 373–391. G. Dangelmayr, E. Knobloch, The Takens–Bogdanov bifurcation with O(2)-symmetry, Phil. Trans. R. Soc. Lond. A 322 (1987) 243–279. J.W. Evans, Nerve axon equations. IV. The stable and unstable impulse, Ind. Univ. Math. J. 24 (1975) 1169–1190. C.K.R.T. Jones, Stability of the travelling wave solution to the FitzHugh–Nagumo equation, Trans. AMS 286 (1984) 431–469. E. Yanagida, Stability of fast traveling pulse solutions of the FitzHugh–Nagumo equations, J. Math. Biol. 22 (1985) 81–104. J. Alexander, R. Gardner, C. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410 (1990) 167–212. R. Pego, M. Weinstein, Eigenvalues, and instabilities of solitary waves, Phil. Trans. R. Soc. Lond. A 340 (1992) 47–94. J. Swinton, The stability of homoclinic pulses: a generalisation of Evans’s method, Phys. Lett. A 163 (1992) 57–62. T.J. Brigdes, G. Derks, Hodge duality and the Evans function, Phys. Lett. A 251 (1999) 363–372. R.A. Gardner, K. Zumbrun, The Gap Lemma and geometric criteria for instability of viscous shock profiles, Commun. Pure Appl. Math. LI (1998) 0797–0855. T. Kapitula, B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Physica D 124 (1998) 58–103. B. Sandstede, Stability of travelling waves, in: B. Fiedler (Ed.), Handbook of Dynamical Systems. II. Towards Applications, Elsevier, Amsterdam, 2002, pp. 983–1055. J.C. Alexander, C.K.R.T. Jones, Existence and stability of asymptotically oscillatory double pulses, J. Reine Angew. Math. 446 (1994) 49–79. J.C. Alexander, C.K.R.T. Jones, Existence and stability of asymptotically oscillatory triple pulses, Z. Angew. Math. Phys. 44 (1993) 189–200. T. Kapitula, The Evans function and generalized Melnikov integrals, SIAM J. Math. Anal. 30 (1998) 273–297. T. Kapitula, Stability criterion for bright solitary waves of the perturbed cubic–quintic Schrödinger equation, Physica D 116 (1998) 95–120. T. Kapitula, B. Sandstede, Edge bifurcations for near integrable systems via Evans function techniques, SIAM J. Math. Anal. 33 (2002) 1117–1143. T. Kapitula, J. Rubin, Existence and stability of standing hole solutions to complex Ginzburg–Landau equations, Nonlinearity 13 (2000) 77–112.
124
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
[20] T.J. Bridges, G. Derks, Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry, Proc. R. Soc. Lond. A 455 (1999) 2427–2469. [21] T.J. Bridges, G. Derks, The symplectic Evans matrix, and the instability of solitary waves and fronts, Arch. Rat. Mech. Anal. 156 (2001) 1–87. [22] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry: II, J. Funct. Anal. 94 (1990) 308–348. [23] P. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1993. [24] E. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955, p. 104. [25] Y.A. Li, K. Promislow, The mechanism of the polarizational mode instability in birefringent fiber optics, SIAM J. Math. Anal. 31 (2000) 1351–1373. [26] A. Goriely, M. Nizette, M. Tabor, On the dynamics of elastic strips, J. Nonlinear Sci. 11 (2001) 3–45.
Instability of local deformations of an elastic rod S. Lafortune∗ , J. Lega Department of Mathematics, University of Arizona, P.O. Box 210089, Tucson, AZ 85721-0089, USA Received 31 May 2002; received in revised form 14 March 2003; accepted 26 March 2003 Communicated by C.K.R.T. Jones
Abstract We study the instability of pulse solutions of two coupled non-linear Klein–Gordon equations by means of Evans function techniques. The system of coupled Klein–Gordon equations considered here describes the near-threshold dynamics of a three-dimensional elastic rod with circular cross-section, subject to constant twist. We determine a condition on the speed of the traveling pulse which ensures spectral instability. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Elastic rod; Klein–Gordon equations; Evans function
1. Introduction When subject to sufficiently high constant twist, elastic rods kept under tension tend to assume a helical shape. Near this bifurcation threshold, the local deformation and twist of the filament can be described in terms of envelope equations for the slowly varying (complex) amplitude of the helical mode and slowly varying twist. Such envelope equations were derived by Goriely and Tabor [1] in the case of filaments with circular cross-section, and take the form of two coupled non-linear Klein–Gordon equations. Lega and Goriely gave a complete classification of traveling wave solutions to this system [2] (see also [3]) and considered the stability of some special solutions, such as traveling holes and stationary periodic structures [2]. A particular family of solutions found in [2] consists of traveling pulses, which correspond to a localized helical structure propagating at constant speed on an otherwise undeformed filament. A reconstruction of such a pulse solution is shown in the numerical simulation of Fig. 1. As illustrated in this figure, reproduced from [2], numerical studies indicate that some pulse solutions are able to propagate in a stable fashion. From the numerics, it is also clear that not all pulses are stable. The goal of this paper is to find a criterion on the speed of the pulse solution which ensures instability. More precisely, we identify a range of speeds for which the point spectrum of the linearized operator about a pulse solution has an eigenvalue on the positive real axis. To this end, we analyze the behavior of the Evans function for the linearized operator near the origin and at infinity, and use a parity argument to conclude that, in some parameter regime, this function vanishes on the positive real axis. ∗
Corresponding author. E-mail addresses: [email protected] (S. Lafortune), [email protected] (J. Lega). 0167-2789/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-2789(03)00125-8
104
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
Fig. 1. Numerical simulation (reproduced from [2]) of the non-linear Klein–Gordon equations (2.1), showing the stable propagation of a traveling pulse solution (given by Eqs. (2.2) and (2.4)). Each panel shows a reconstruction of the corresponding elastic filament. Parameters used in the simulation are ω = 0, µ = −864 and c02 = 2.4. The speed of the pulse is c 0.6124, and the box length is 1 dimensionless unit.
The Evans function [4–7] of a linear operator acting on functions defined on the real line is an analytic function on some region of the complex plane which vanishes on the point spectrum of the operator, and which takes on real values on the real axis. It can be expressed as a scattering coefficient [6], as a determinant whose entries involve inner products of an eigenfunction of the linear operator with an eigenfunction of its adjoint operator [4,5,8–10], or as a Wronskian [7,10–13]. Applications of this technique include stability results for the propagation of an action-potential in nerve axon equations [4] and in the FitzHugh–Nagumo equations [5–7], for pulse solutions to the generalized Korteweg–de Vries, Benjamin–Bona–Mahoney and Boussinesq equations [8], for multi-pulse solutions to reaction–diffusion equations [14,15], for solutions to perturbed non-linear Schrödinger equations [12,16,17] and to near integrable systems [18], and for traveling hole solutions of the one-dimensional complex Ginzburg–Landau equation near the non-linear-Schrödinger limit [19]. The point of view we adopt here is that reviewed by Sandstede [13], where the Evans function is defined as a Wronskian, which vanishes if initial conditions giving convergence at plus and minus infinity are linearly dependent. Because it is often difficult or even impossible to calculate the Evans function for all complex values of the spectral parameter, instability of a solution to a system of partial differential equations is typically proven by showing that the Evans function of the associated linearized operator has to vanish at least once on the positive real axis. More precisely, knowledge of the behavior of the Evans function near the origin and for real and large values of the spectral parameter are often sufficient to conclude that, by continuity, the Evans function should vanish for some real positive value of the spectral parameter. This argument is classical (see for instance [4–6,8,11,14,15,20,21]), and it is also what we use in this paper. The calculations presented here however differ from standard applications of Evans function techniques for the following reasons. First, the linear system we are interested in has three-dimensional stable and unstable eigenspaces at ±∞, which we have to track carefully as the spectral parameter goes from the origin to infinity. Second, we will see that the first four derivatives of the Evans function vanish at the origin. In order to calculate the fifth derivative, we look for expansions of the eigenfunctions of the linear operator in powers of the spectral parameter. But because the asymptotic linear operator has a marginal mode which is bounded but
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
105
non-decaying at infinity, both regular and singular expansions are needed. The existence of a zero of order five of the Evans function may seem surprising for a Hamiltonian system [22], but we will see that another particularity of this problem is that the eigenvalue zero is embedded in the continuous spectrum. This paper is organized as follows. The coupled Klein–Gordon equations and their one-parameter family of pulse solutions are introduced in Section 2. Section 3 is devoted to a discussion of the properties of the linearized system. In particular, we are able to calculate a fundamental matrix describing the solutions of the six-dimensional first order system obtained when the spectral parameter is zero. In Section 4, we develop regular and singular expansions of the eigenfunctions in powers of the spectral parameter. In Section 5, we define the Evans function, calculate its fifth derivative at the origin, and determine its behavior for large values of the spectral parameter. The instability criterion is given at the end of Section 5. Conclusions and open questions are discussed in Section 6.
2. The coupled Klein–Gordon equations The near-threshold dynamics of an elastic rod with circular cross-section is described by the following dimensionless coupled Klein–Gordon equations [1] 2 ∂2 A ∂B 2∂ A − c = µA − A|A|2 + A , 0 2 2 ∂x ∂t ∂x
∂2 B ∂2 B ∂|A|2 − = − , ∂x ∂t 2 ∂x2
(2.1)
where A(x, t) represents the (scaled) slowly varying complex amplitude of the helical mode which grows above the bifurcation threshold, B(x, t) is the (scaled) real axial twist, c0 and µ are real constants. The pulse solutions we are interested in belong to a class of solutions of the form A = a(ξ) eiωt ,
B = b(ξ),
ξ = x − ct,
(2.2)
where a and b solve the following ordinary differential equations in ξ (c2 − c02 )a − 2icωa = a(ω2 + µ − |a|2 + b ),
(c2 − 1)b = −(|a|2 ) .
(2.3)
As discussed in [2] (see also [3]), traveling pulses form a two-parameter family of solutions to (2.1), given by ωc α2 a0 (ξ) = α sech(βξ) exp i ξ , b (ξ) = tanh (βξ), (2.4) 0 β(1 − c2 ) c2 − c02 where α2 =
2(c2 − 1) (µ(c2 − c02 ) − ω2 c02 ), c2 (c2 − c02 )
β2 =
µ(c2 − c02 ) − ω2 c02 (c2 − c02 )2
.
(2.5)
In Section 3, we will restrict ourselves to the case ω = 0, which corresponds to “non-rotating” boundary conditions. Conditions (2.5) can then be simplified into α2 =
2µ 2 (c − 1), c2
β2 =
c2
µ . − c02
The linearization of the Klein–Gordon equations (2.1) about A = 0 and B constant shows that amplitude perturbations of the form exp[i(kx + Ωt)] have a dispersion relation given by −Ω2 = µ − c02 k2 . This indicates that when µ > 0, Fourier modes with wave number k such that k2 < µ/c02 may experience growth. As a consequence, when µ > 0, the asymptotic state of the pulse solution is unstable, since the continuous spectrum of the linearization
106
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
about the pulse solution intersects the right half-plane. In what follows, we therefore only consider the case µ < 0. The following conditions should then be satisfied in order for α and β to be real c2 < c02 ,
−1 < c < 1.
(2.6)
For convenience, we will also assume that β and c0 are positive. There is no loss of generality in doing so since the pulse solution (2.4) is not affected by changing the signs of β or c0 .
3. The linear problem We start by linearizing (2.1) around the pulse solution (2.4) for ω = 0. We write the perturbed solution as A(x, t) = a0 (ξ) + u(ξ, ˜ t),
¯ A(x, t) = a¯ 0 (ξ) + v˜ (ξ, t),
B(x, t) = b0 (ξ) + w(ξ, ˜ t),
where u, ˜ v˜ and w ˜ are small. Substitution of these expressions in Eq. (2.1) and linearization give rise to a system of the form I3 O3 ∂Y , ˜ V˜ , W) ˜ T, L = = LY, Y = (u, ˜ v˜ , w, ˜ U, ˜ 2cI3 ∂ ∂t L ∂ξ ˜ is given by where O3 and I3 are the 3 × 3 zero and identity matrices, respectively and the 3 × 3 operator L 2 ∂ 2 2 ∂ −a02 (ξ) a0 (ξ) (c0 − c ) ∂ξ 2 + C0 ∂ξ 2 ∂ ∂ 2 2 2 , C0 = µ − 2|a0 (ξ)|2 + b (ξ). ˜ L= −¯a0 (ξ) (c0 − c ) 2 + C0 a¯ 0 (ξ) 0 ∂ξ ∂ξ ∂ ∂ ∂2 − a¯ 0 (ξ) + a¯ 0 (ξ) − a0 (ξ) + a0 (ξ) (1 − c2 ) 2 ∂ξ ∂ξ ∂ξ Since b0 is finite at ±∞, we allow perturbations of B which are bounded (as opposed to decaying to zero) on the real line. We therefore consider the spectral problem λY = LY , where Y is given above and (u, ˜ v˜ , w) ˜ = (u(ξ), v(ξ), w(ξ)) ∈ (H 1 × H 1 × L∞ ) ∩ (C1 × C1 × C1 ).
(3.1)
The functions u, v and w then satisfy a six-dimensional linear dynamical system of the form X = A(ξ, λ)X,
X = (u, v, w, u , v , w )T ,
where A is the 6 × 6 matrix given by 0 0 0 0 0 0 a02 (ξ) C1 − 2 2 c2 − c02 A(ξ, λ) = c − c0 2 C1 − a¯ 0 (ξ) c2 − c2 2 c − c02 0 a¯ (ξ) a (ξ) − 20 − 20 c −1 c −1
0 0 0 0
(3.2)
1 0 0 2cλ c2 − c02
0 −
λ2 c2 − 1
0 −
a¯ 0 (ξ) c2 − 1
0 1 0 0 2cλ − c02
c2 −
a0 (ξ) c2 − 1
0 0 1 a0 (ξ) c2 − c02
, a¯ 0 (ξ) c2 − c02 2cλ c2 − 1
(3.3)
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
107
with C1 = −λ2 + µ − 2|a0 (ξ)|2 + b0 (ξ) = −λ2 + C0 . We are interested in identifying bounded solutions to (3.2) for real positive values of the spectral parameter λ. When they exist, such solutions will be called eigenfunctions of the linear operator L, and the corresponding values of the spectral parameter λ will be called eigenvalues. Linear system (3.2) has non-constant coefficients and, as far as we know, cannot be solved explicitly for arbitrary values of λ. We first look at the particular case λ = 0, which can be solved completely, and second at the asymptotic system obtained by taking the limit of (3.2) as ξ → ±∞. We then discuss under what conditions the limit as ξ → ±∞ of solutions to (3.2) are solutions of this asymptotic system. 3.1. Solution to the linear problem for λ = 0 In the case ω = 0, symmetry properties of (2.3) allow us to completely solve the system X = A(ξ, 0)X,
(3.4)
where A(ξ, 0) is the matrix given by (3.3) with λ = 0. It can indeed be shown, using a well known algorithm [23], that there are only four Lie point symmetries for system (2.3). These symmetries are reductions of the symmetries of the system of PDEs (2.1) and are listed below. We use the following notation: a(ξ) and b(ξ) are solutions of ˜ system (2.3) and the new solutions a˜ (ξ) and b(ξ) are obtained by applying the symmetries to a and b. 1. Reference frame invariance: ˜ b(ξ) = b(ξ + ξ0 ),
a˜ (ξ) = a(ξ + ξ0 ),
(3.5)
where ξ0 is a real arbitrary constant. 2. Uniform shift on b: a˜ (ξ) = a(ξ),
˜ b(ξ) = b(ξ) + χ,
(3.6)
where χ is a real arbitrary constant. 3. Gauge invariance: ˜ b(ξ) = b(ξ),
a˜ (ξ) = a(ξ) eiφ ,
(3.7)
where φ is a real arbitrary constant. 4. Dilation invariance (present only when ω = 0): a˜ (ξ) = γa(γξ),
˜ b(ξ) = γb(γξ) + µ(γ 2 − 1)ξ,
(3.8)
where γ is a real constant. Each of these symmetries gives us a solution to (3.4). Indeed, consider an analytic symmetry with parameter ( ∈ R, whose action on the solution (a0 (ξ), b0 (ξ)) of (2.3) is given by a˜ 0( (ξ) = Γ1 (ξ, () = a0 (ξ) + (φ1 (ξ) + · · · ,
b˜ 0( (ξ) = Γ2 (ξ, () = b0 (ξ) + (φ2 (ξ) + · · · .
(3.9)
Since (˜a0( (ξ), b˜ 0( (ξ)) is also a solution of (2.3), the terms in (3.9) linear in ( must be a solution of the linearization of (2.3) around the solution (a0 (ξ), b0 (ξ)), i.e. (φ1 (ξ), φ2 (ξ)) must solve Eq. (3.4). Since we restricted ourselves to the case ω = 0, Eq. (2.3) then has four independent symmetries (3.5)–(3.8), which give us four linearly independent solutions to the linear system (3.4). These solutions are listed below (we
108
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
only write the first three components of the vector X; the last three components are the derivatives of the components shown): 0 X1 (ξ) = 0 , 1
a0 (ξ) X2 (ξ) = −a0 (ξ) , 0
a0 (ξ)
X3 (ξ) = a0 (ξ) , b0 (ξ)
X4 (ξ) =
a0 (ξ) + ξa0 (ξ) a0 (ξ) + ξa0 (ξ) b0 (ξ) + ξb0 (ξ) + 2µξ
. (3.10)
Note that in the limit ξ → ±∞, the solutions X2 and X3 converge to zero, X1 is constant and X4 diverges. In other words, only three of the four symmetries give solutions which are bounded at infinity. Note also that since ω = 0, a0 (ξ) in (2.4) is a real solution and thus a¯ 0 (ξ) = a0 (ξ). It turns out that the two remaining solutions in a fundamental set of solutions of (3.4) can be found by reduction of order and are given by
sinh (2βξ) ξ (ξ) + a0 4β 2 sinh (2βξ) ξ , X5 (ξ) = + −a0 (ξ) 4β 2
0
X6 (ξ) =
a0 (ξ)
sinh (2βξ) ξ + 16β 8
sinh (2βξ) ξ + 16β 8 sinh (2βξ) ξ ξµ b0 (ξ) + + 2 (1 − c2 ) 16β 8 2c a0 (ξ)
. (3.11)
These solutions diverge as ξ → ±∞. Since the linearized system at λ = 0 has only three linearly independent solutions in the class of admissible perturbations defined by (3.1), λ = 0 is an eigenvalue of L of geometric multiplicity equal to 3. In the same way as symmetries of the equations give genuine eigenfunctions, symmetries of a family of solutions can also lead to generalized eigenfunctions. More precisely, pulse solutions to (2.1), written as a first order system, ∂Y/∂t = F(Y), are of the form G(ωt)T(−ct)ψc,ω (x), where T corresponds to translational invariance (T(ct)ϕ(x) = ϕ(x + ct)), G to gauge invariance (G(ωt)ϕ(x) = exp(iωD0 t)ϕ(x), where D0 = diag(1, −1, 0, 1, −1, 0)) and ψc,ω (x) is a six component function of x parametrized by c and ω. When such a solution is substituted into (2.1), we have ∂ [G(ωt)T(−ct)ψc,ω (x)] = F(G(ωt)T(−ct)ψc,ω (x)) = G(ωt)F(T(−ct)ψc,ω (x)), ∂t i.e. iωD0 G(ωt)T(−ct)ψc,ω (x) − cG(ωt)
∂ [T(−ct)ψc,ω (x)] = G(ωt)F(T(−ct)ψc,ω (x)), ∂ξ
ξ = x − ct.
Since the action of G is multiplicative, the above equation can be re-written as iωD0 T(−ct)ψc,ω (x) − c
∂ [T(−ct)ψc,ω (x)] = F(T(−ct)ψc,ω (x)). ∂ξ
(3.12)
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
109
By taking the derivative of (3.12) with respect to c we get ∂ ∂ ∂ ∂ iωD0 T(−ct) ψc,ω (x) − [T(−ct)ψc,ω (x)] − c T(−ct) ψc,ω (x) ∂ξ ∂c ∂ξ ∂c ∂ = DF(T(−ct)ψc,ω (x)) T(−ct) ψc,ω (x) , ∂c where DF is the Jacobian of F . When we set ω = 0, the above equation implies ∂ ∂ P = − P, L ∂c ∂ξ where P is the pulse solution at ω = 0. Since the right-hand side is in the kernel of L, ∂P/∂c is a generalized eigenfunction of L belonging to the eigenvalue λ = 0. Similarly, by taking the derivative of (3.12) with respect to ω and setting ω = 0, one finds that the derivative of the pulse solution with respect to ω, taken at ω = 0, Pω is a generalized eigenfunction of L such that L(Pω ) = iD0 P. Therefore, λ = 0 is an eigenvalue of L of algebraic multiplicity 5 when ω = 0. When ω = 0, one can still solve the linear problem at λ = 0 as follows. The three symmetries (3.5)–(3.7) still give three genuine eigenfunctions. Three other linearly independent solutions can be found by variation of parameters. It turns out that the system of equations for the unknown functions of ξ decouples into a two-by-two first order system and a first order linear differential equation. The latter equation can always be solved, at least formally, and, by rescaling the independent variable and one of the dependent variables in the two-by-two system, one obtains the same two-by-two system as in the case ω = 0, which can thus also be solved. The formula for a fundamental matrix of the linear differential system in ξ is however very cumbersome and it is only when ω = 0 that the calculations presented below are tractable. The above discussion for the existence of two generalized eigenfunctions for λ = 0 also applies when ω = 0. This argument is similar to that of [22] for Hamiltonian symmetries, but is more generic in the sense that it should work for any two-parameter family of solutions of a partial differential equation for which the two parameters are due to translational and gauge invariance. 3.2. Asymptotic system We now turn to the asymptotic system obtained as ξ → ±∞. Let A∞ (λ) = lim A(ξ, λ), ξ→±∞
where A is given by (3.3). Since a0 (ξ) and b0 (ξ) both converge to zero as ξ → ±∞, we have 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 µ − λ2 2cλ 0 0 0 0 2 − c2 A∞ (λ) = c2 − c2 . c 0 0 2 2cλ µ − λ 0 0 0 0 2 2 2 2 c − c0 c − c0 2 λ 2cλ 0 0 − 2 0 0 c −1 c2 − 1
110
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
The eigenvalues νi∞ (λ) and corresponding eigenvectors v∞ i,j (λ) of this matrix can be calculated explicitly. Its eigenvalues are given by
cλ + µ(c2 − c02 ) + c02 λ2 λ λ ν1∞ (λ) = , ν2∞ (λ) = , ν3∞ (λ) = , c+1 c−1 c2 − c02
cλ − µ(c2 − c02 ) + c02 λ2 , (3.13) ν4∞ (λ) = c2 − c02 where ν3∞ (λ) and ν4∞ (λ) have geometric multiplicity 2. The corresponding eigenvectors are ∞ v1 (λ) =
0 0 1 0 0 ν1∞ (λ)
v∞ (λ) = 3,2 ∞ v4,2 (λ) =
∞ v2 (λ) =
,
(c2 − c02 )ν3∞ (λ) − 2cλ 0 0 µ − λ2 0 0 (c2 − c02 )ν4∞ (λ) − 2cλ 0 0 µ − λ2 0 0
,
0
,
0 1 0 0 ν2∞ (λ)
0
2 (c − c2 )ν∞ (λ) − 2cλ 0 3 0 ∞ , v3,1 (λ) = 0 2 µ−λ 0
0
(c2 − c2 )ν∞ (λ) − 2cλ 0 4 0 , v∞ (λ) = 4,1 0 µ − λ2 0
.
(3.14)
The convergence, as ξ → ±∞, of solutions to X = A∞ (λ)X depends on the real parts of the eigenvalues νi∞ (λ). If the real and imaginary parts of λ are denoted by λr and λi , then
√ 2cλ + R + µ(c2 − c02 ) + c02 (λ2r − λ2i ) r λ λ r r , Re(ν1∞ ) = , Re(ν2∞ ) = , Re(ν3∞ ) = √ c+1 c−1 2(c2 − c02 )
√ 2cλr − R + µ(c2 − c02 ) + c02 (λ2r − λ2i ) Re(ν4∞ ) = , √ 2(c2 − c02 ) where R=
(µ(c2 − c02 ) + c02 (λ2r − λ2i ))2 + 4c04 λ2r λ2i .
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
111
It is a simple matter of calculation to prove that the only location in the complex plane where at least one of the Re(νi∞ ) may vanish is on the imaginary axis (i.e. where λr = 0). Moreover, for λr = 0 and |λi | <
µ(c2 − c02 )/c02 ,
only two eigenvalues, ν1∞ and ν2∞ , are purely imaginary (recall that µ(c2 − c02 ) > 0 because µ < 0 and because of
(2.6)), whereas all four eigenvalues are imaginary when λr = 0 and |λi | ≥ µ(c2 − c02 )/c02 . Since the eigenvalues νi∞ are continuous functions of λ, the only place where their real part may change sign is therefore when λ crosses the imaginary axis. When λ is real and sufficiently large, the real parts of ν1∞ and ν4∞ are positive and the real parts of ν2∞ and ν3∞ are negative. Similarly, when λ is a sufficiently large real negative number, the real parts of ν2∞ and ν4∞ are positive and the real parts of ν1∞ and ν3∞ are negative. Together with the fact that Re(νi∞ ) can only change sign when λ is on the imaginary axis, these statements imply that A∞ (λ) has three eigenvalues (counting multiplicity) with strictly positive real parts and three eigenvalues with strictly negative real parts for all λ’s with non-zero real part. In the region of interest, i.e. in the right half complex plane (where λr ≥ 0), the situation can be summarized as follows: Re(ν1∞ (λ)) ≥ 0,
Re(ν2∞ (λ)) ≤ 0,
Re(ν3∞ (λ)) ≤ 0,
Re(ν4∞ (λ)) ≥ 0.
Following [8], we define the deviator to be R(ξ, λ) = A(ξ, λ) − A∞ (λ). Since the components of the matrix A(ξ, λ) converge exponentially to the components of A∞ (λ) as ξ → ±∞, it is clear that the integral ∞ |R(ξ, λ)| dx, (3.15) −∞
converges uniformly in λ on any compact subset of C. We now turn to solutions of (3.2) which are bounded either as ξ → ∞ or as ξ → −∞. It is known [24] that if the integral (3.15) converges and if the matrix A∞ (λ) is diagonalizable (although this last condition is not necessary; see [7]), solutions to (3.2) that are bounded for ξ positive behave, in the limit as ξ → ∞, like solutions of the constant coefficient system X = A∞ (λ)X. More precisely, when Re(λ) > 0 the matrix A∞ (λ) has a three-dimensional eigenspace corresponding to eigenvalues with negative real parts, and hence there is a three-dimensional space of solutions to (3.2) which are bounded as ξ → ∞. Moreover, three linearly independent solutions φi (ξ, λ), i = 1, 2, 3, to (3.2) can be chosen in this space such that ∞ (λ)ξ
lim φ1 (ξ, λ) e−ν2
ξ→∞
= v∞ 2 (λ),
∞ (λ)ξ
lim φ2 (ξ, λ) e−ν3
ξ→∞
= w2 (λ),
∞ (λ)ξ
lim φ3 (ξ, λ) e−ν3
ξ→∞
= w3 (λ), (3.16)
where w2 and w3 are any two linearly independent vectors in the eigenspace of A∞ (λ) belonging to ν3∞ (λ). A similar result holds for solutions to (3.2) which are bounded as ξ → −∞. Three such solutions φi (ξ, λ), i = 4, 5, 6, can then be chosen such that ∞ (λ)ξ
lim φ4 (ξ, λ) e−ν1
ξ→−∞
= v∞ 1 (λ),
∞ (λ)ξ
lim φ5 (ξ, λ) e−ν4
ξ→−∞
= w5 (λ),
∞ (λ)ξ
lim φ6 (ξ, λ) e−ν4
ξ→−∞
= w6 (λ), (3.17)
∞ where w5 and w6 are any two linearly independent vectors in the eigenspace generated by v∞ 4,1 (λ) and v4,2 (λ). Regarding the analyticity of the φi ’s, one must first observe that there exists a domain containing the positive real
112
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
axis λr > 0, in which the eigenvectors (3.14) and eigenvalues (3.13) are analytic and in which the number of eigenvalues with distinct real parts does not change. Because of this and because the convergence of the integral (3.15) is uniform in λ on any compact subset of C, the φi are analytic in λ ∈ R, λ > 0, for each fixed ξ [24]. When λ = 0, the eigenvalues ν1∞ (λ) (ν1∞ (λ) ≥ 0 for λ ∈ R+ ) and ν2∞ (λ) (ν2∞ (λ) ≤ 0 for λ ∈ R+ ) vanish. The above argument can be extended to show that there exist solutions φi whose asymptotic behavior is given by (3.16) and (3.17) and which are analytic in λ in a neighborhood of λ = 0 if R(ξ, λ) = O(exp(−γ|ξ|))
as |ξ| → ∞, γ > 0,
and if γ > |Re(ν1∞ (λ) − ν2∞ (λ))|. Such solutions are constructed in the next section. This illustrates how the Evans function can be extended in an analytic fashion to values of the spectral parameter for which the asymptotic matrix has eigenvalues with zero real part (in the L2 case, this occurs when the spectral parameter enters the continuous spectrum). Analytic continuation of the Evans function across the continuous spectrum has been the topic of several recent papers (see for instance [11,12,19,25]) and is addressed by the Gap Lemma [11,12]. The above argument, together with the calculations of Section 4, provides an alternative proof of the Gap Lemma in the case where A∞ (λ) is diagonalizable for Re(λ) = 0. We also note that since the eigenvalues of A∞ (λ) are analytic in λ in a neighborhood of λ = 0, the Evans function defined in Section 5.1 does not have a branch point at the origin. The reader is referred to [19] for a situation where an eigenvalue lies on a branch point of the Evans function. Finally, since the matrix A in (3.3) is real when λ is real, the φi can also be chosen to be real in this particular case.
4. Asymptotic expansions of solutions for small values of the spectral parameter This section is devoted to the behavior, for small real λ, of the solutions φi (ξ, λ) defined above. We know a fundamental matrix for the system X = A(ξ, 0)X, obtained when λ = 0, and we look for asymptotic expansions of the φi ’s as λ → 0, which are uniformly valid in ξ. Regular expansions of the form (0)
(1)
(2)
φi (ξ, λ) = φi (ξ) + φi (ξ)λ + φi (ξ)λ2 + · · · ,
(4.1)
are sought for φ2 , φ3 (for ξ ≥ 0), φ5 and φ6 (for ξ ≤ 0) since the corresponding eigenvalues νi∞ (λ) have non-zero real parts when λ = 0, whereas singular expansions are sought for φ1 and φ4 . The latter are of the form ∞ (λ)ξ
φ1 (ξ, λ) = eν2
∞ (λ)ξ
φ4 (ξ, λ) = eν1
(0) (1) (2) (φ˜ 1 (ξ) + φ˜ 1 (ξ)λ + φ˜ 1 (ξ)λ2 + · · · ), (0)
(1)
(2)
(φ˜ 4 (ξ) + φ˜ 4 (ξ)λ + φ˜ 4 (ξ)λ2 + · · · ),
(4.2)
(j) (j) (j) where the φ˜ i are such that limξ→∞ φ˜ 1 (ξ) and limξ→−∞ φ˜ 4 (ξ) exist and are finite (see also [12,19] for singular (0) (0) expansions near a branch point of the Evans function). The φi and φ˜ i satisfy the linear homogeneous system (j) (j) (3.4). The higher order terms φi (resp. φ˜ i ) with j > 0 solve a linear system of the form X − A(ξ, 0)X = f(ξ), (k) (k) where the inhomogeneous part is determined by terms found at preceding orders, i.e. by the φi (resp. φ˜ i ) with k < j. Since the homogeneous system can be solved completely and has solutions given by (3.10) and (3.11), the inhomogeneous systems obtained at each order can be solved by the method of variation of parameters. (0) (0) At order zero, the φi and φ˜ i are linear combinations of the Xi ’s. One can check that ∞ lim X1 (ξ) = v∞ 1 (0) = v2 (0) = e3 ,
ξ→±∞
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
113
where e3 = (0, 0, 1, 0, 0, 0)T and that X2 and X3 satisfy 2α ∞ (v∞ 3,1 (0) − v3,2 (0)), ξ→∞ − c02 ) 2α ∞ ∞ (v∞ lim X3 (ξ) e−ν3 (0)ξ = 3,1 (0) + v3,2 (0)), ξ→∞ c2 − c02 2α ∞ ∞ lim X2 (ξ) e−ν4 (0)ξ = (−v∞ 4,1 (0) + v4,2 (0)), 2 ξ→−∞ β(c − c02 ) 2α ∞ ∞ lim X3 (ξ) e−ν4 (0)ξ = (v∞ 4,1 (0) + v4,2 (0)). 2 ξ→−∞ c − c02 ∞ (0)ξ
lim X2 (ξ) e−ν3
=
β(c2
It is thus possible to choose the first terms in the expansions (4.1) and (4.2) to be (0) (0) φ˜ 1 (ξ) = φ˜ 4 (ξ) = X1 (ξ),
(0)
(0)
(0)
φ2 (ξ) = φ5 (ξ) = X2 (ξ),
(0)
φ3 (ξ) = φ6 (ξ) = X3 (ξ),
(4.3)
with the wi in (3.16) and (3.17) then given by 2α ∞ (v∞ 3,1 (λ) − v3,2 (λ)), β(c2 − c02 ) 2α ∞ w5 (λ) = (−v∞ 4,1 (λ) + v4,2 (λ)), 2 β(c − c02 )
w2 (λ) =
2α ∞ (v∞ 3,1 (λ) + v3,2 (λ)), c2 − c02 2α ∞ w6 (λ) = (v∞ 4,1 (λ) + v4,2 (λ)). 2 c − c02
w3 (λ) =
(4.4)
(1)
We now look at first order corrections. As mentioned before, the φi satisfy inhomogeneous equations that can be (0) (1) solved once the φi are chosen. In the case of φ2 , the most general solution of the corresponding inhomogeneous linear equation which decays to zero as ξ → ∞ is given by (1)
(1)
φ2 (ξ) = φ2,part (ξ) + c2,2 X2 (ξ) + c2,3 X3 (ξ),
(4.5) (1)
where c2,2 and c2,3 are arbitrary real constants and φ2,part (ξ) is a particular solution found using variation of constants αc(2βξ + 1) sech(βξ) 2β(c2 − c02 ) αc(2βξ + 1) sech(βξ) (1) φ2,part (ξ) = − . 2β(c2 − c02 ) 0
(4.6)
As before, we only show the first three components of each vector. The constants c2,2 and c2,3 are determined by condition (3.16) with w2 given in (4.4). The behavior near the origin of the Evans function defined in the next section (1) will depend on φ2,part (ξ) but will not be affected by the particular values of c2,2 and c2,3 . We will be only interested in finding the sign of the lowest non-zero derivative of the determinant (5.1) at λ = 0; adding linear combinations (j) (0) of the φi to the φi with j > 0 and i = 2, 3, 5, 6 only alters higher order derivatives. We therefore do not evaluate c2,2 and c2,3 . (1) Similarly, φ3 can be determined to be (1)
(1)
φ3 (ξ) = φ3,part (ξ) + c3,2 X2 (ξ) + c3,3 X3 (ξ),
(4.7)
114
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
where
α(c2 (1 − c2 )(1 + 2βξ) sinh (βξ) + 2(c02 − c2 ) e−βξ )
2c cosh 2 (βξ)(c2 − 1)(c2 − c02 ) α(c2 (1 − c2 )(1 + 2βξ) sinh (βξ) + 2(c02 − c2 ) e−βξ ) (1) φ3,part (ξ) = 2c cosh 2 (βξ)(c2 − 1)(c2 − c02 ) µ((c2 − 1)(c2 − 2c02 ) e−2βξ + 2c2 (1 − c02 + βξ(1 − c2 ))) βc3 cosh 2 (βξ)(c2 − 1)(c2 − c02 )
.
(4.8)
The first order terms in φ5 (ξ, λ) and φ6 (ξ, λ) must converge to zero as ξ → −∞, i.e. (j)
lim φi (ξ) = 0,
ξ→−∞
i = 5, 6,
(4.9)
and are determined in a similar fashion. We obtain (1)
(1)
(1)
(1)
φ5 (ξ) = φ2,part (ξ) + c5,2 X2 (ξ) + c5,3 X3 (ξ),
(4.10)
and φ6 (ξ) = φ3,part (ξ) − 4
µ(c2 − 2c02 ) βc3 (c2 − c02 )
X1 (ξ) + c6,2 X2 (ξ) + c6,3 X3 (ξ),
(4.11) (1)
where the term in X1 was added to make sure that condition (4.9) is satisfied by φ6 . Recall that X1 , given in (3.10) (1) (1) is simply the constant vector e3 . Finally, the functions φ˜ 1 (ξ) and φ˜ 4 (ξ) are found to be (1) φ˜ 1 (ξ) = −
ξ X4 (ξ) e3 + + c1,1 X1 (ξ) + c1,2 X2 (ξ) + c1,3 X3 (ξ), c−1 2µ(c − 1)
where X4 is given in (3.10) and ξ X4 (ξ) (1) φ˜ 4 (ξ) = − e3 + + c4,1 X1 (ξ) + c4,2 X2 (ξ) + c4,3 X3 (ξ). c+1 2µ(c + 1) (i)
(i)
In order to have more uniform notation, we also give expressions for φ1 (ξ) and φ4 (ξ) obtained from (4.2) by expanding the exponential in powers of λ. We have (0) (0) φ1 (ξ) = φ˜ 1 (ξ),
(0) (0) φ4 (ξ) = φ˜ 4 (ξ), ξ ˜ (0) X4 (ξ) (1) (1) φ1 (ξ) = φ˜ 1 (ξ) + φ1 (ξ) = + c1,1 X1 (ξ) + c1,2 X2 (ξ) + c1,3 X3 (ξ), c−1 2µ(c − 1) X4 (ξ) (1) φ4 (ξ) = + c4,1 X1 (ξ) + c4,2 X2 (ξ) + c4,3 X3 (ξ). 2µ(c + 1)
(4.12)
Terms at order two are given in Appendix A. We will not need to go any further in the asymptotic expansions to be
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
115
able to characterize the behavior of the Evans function near the origin. To summarize, we have X4 (ξ) λ+ c1,i Xi (ξ)λ + O(λ2 ), 2µ(c − 1) 3
φ1 (ξ, λ) = X1 (ξ) +
i=1
(1)
φ2 (ξ, λ) = X2 (ξ) + φ2,part (ξ)λ +
3
(2)
c2,i Xi (ξ)λ + φ2,part (ξ)λ2 +
i=2
+
3
3 i=2
(1)
c2,i φi,part (ξ)λ2
e2,i Xi (ξ)λ2 + O(λ3 ),
i=2 (1)
φ3 (ξ, λ) = X3 (ξ) + φ3,part (ξ)λ +
3
(2)
c3,i Xi (ξ)λ + φ3,part (ξ)λ2 +
i=2
+
3
3 i=2
(1)
c3,i φi,part (ξ)λ2
e3,i Xi (ξ)λ2 + O(λ3 ),
i=2
X4 (ξ) c4,i Xi (ξ)λ + O(λ2 ), λ+ 2µ(c + 1) 3
φ4 (ξ, λ) = X1 (ξ) +
i=1
(1)
φ5 (ξ, λ) = X2 (ξ) + φ2,part (ξ)λ +
3
(2)
c5,i Xi (ξ)λ + φ5,part (ξ)λ2
i=2
+
3 i=2
(1)
c5,i φi,part (ξ)λ2 +
φ6 (ξ, λ) = X3 (ξ) + +
3 i=2
(1)
(1) φ3,part (ξ) − 4
(1)
3
e5,i Xi (ξ)λ2 + O(λ3 ),
i=2
µ(c2 − 2c02 ) βc3 (c2
c6,i φi,part (ξ)λ2 +
3
− c02 )
X1 (ξ) λ +
3
(2)
c6,i Xi (ξ)λ + φ6,part (ξ)λ2
i=2
e6,i Xi (ξ)λ2 + O(λ3 ),
(4.13)
i=2 (2)
where the φi,part are given in (4.6) and (4.8) and the φi,part are given in Appendix A. Terms of order two or higher in λ that will not be needed in the next section have been omitted. The sums reflect the fact that, at each order greater than zero, convenient linear combinations of X1 , X2 and X3 should be included in order to ensure that conditions (3.16) and (3.17) are satisfied. However, as mentioned above, the actual values of the ci,j and ei,j are irrelevant since, as shown in Section 5.2, they do not change the value of the first non-zero derivative of the Evans function (5.1) at the origin.
5. Evans function 5.1. Definition We defined eigenvalues as values of λ for which the system (3.2) has a solution which is bounded as ξ → ±∞, as defined by (3.1). Equivalently, λ0 is an eigenvalue if and only if the space of solutions bounded as ξ → +∞, spanned by {φ1 , φ2 , φ3 }, and the space of solutions bounded as ξ → −∞, spanned by {φ4 , φ5 , φ6 }, have an intersection of
116
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
strictly positive dimension when λ = λ0 . This indeed implies that there is a non-trivial solution to (3.2) with λ = λ0 , which is bounded as ξ → ±∞. The most straightforward way to test whether these two spaces of solutions intersect non-trivially is to calculate the determinant of the two spanning sets. Since it suffices to calculate this determinant at ξ = 0, we define ui (λ) = φi (0, λ),
i = 1, 2, . . . , 6.
Then λ0 is an eigenvalue of (3.2) if and only if the function D(λ) = det(u1 (λ), u2 (λ), u3 (λ), u4 (λ), u5 (λ), u6 (λ)),
(5.1)
evaluated at λ = λ0 is zero. The function D(λ) is called the Evans function [4–7,11–13,19,25]. It is analytic in a neighborhood of the positive real axis and it is real for λ real. In what follows, we study the behavior of the Evans function at the origin and for large values of the spectral parameter λ. 5.2. Behavior of the Evans function near the origin In Section 4, we considered expansions of the φi ’s near λ = 0. The functions ui (λ) can also be expanded as (0)
(1)
(2)
ui (λ) = ui + ui λ + ui λ2 + · · · , (j)
(j)
where ui = φi (0). From (4.13), one can see that u5 (λ) − u2 (λ) = λ
3 i=2
(0)
d2,i ui + O(λ2 ),
u6 (λ) − u3 (λ) = λ
3 i=1
(0)
d3,i ui + O(λ2 ),
where d2,i = c5,i − c2,i , d3,i = c6,i − c3,i for i = 2, 3, and d3,1 = −4(µ(c2 − 2c02 ))/(βc3 (c2 − c02 )). One can therefore re-write D(λ) as 3 3 D(λ) = det u1 (λ), u2 (λ), u3 (λ), u4 (λ), u5 (λ) − λ d2,i ui (λ), u6 (λ) − λ d3,i ui (λ) i=2
i=1
≡ det(u1 (λ), u2 (λ), u3 (λ), u4 (λ), u∗5 (λ), u∗6 (λ)),
(5.2)
where u∗5 (λ) − u2 (λ) = O(λ2 ),
u∗6 (λ) − u3 (λ) = O(λ2 ).
(5.3)
The Evans function (5.1) then becomes D(λ) = D(0) + D(1) λ + D(2) λ2 + · · · ,
(5.4) (j)
where each D(n) can be calculated in terms of the ui (i ) (i ) (i ) (i ) ∗(i ) ∗(i ) D(n) = det(u1 1 , u2 2 , u3 3 , u4 4 , u5 5 , u6 6 ),
(5.5)
|i|=n
with i = (i1 , i2 , i3 , i4 , i5 , i6 ), |i| = i1 + i2 + i3 + i4 + i5 + i6 . Because of the equalities given in (4.3), the two (0) (0) (0) (0) ∗(0) ∗(0) spaces spanned by {u1 , u2 , u3 } and by {u4 , u5 , u6 } are identical since (0)
(0)
u1 = u4 ,
(0)
(0)
∗(0)
u2 = u5 = u5
,
(0)
(0)
∗(0)
u3 = u6 = u6
.
(5.6)
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
117
It is thus easy to conclude from (5.6) that at least three of the ij ’s must be non-zero in order for one of the terms in the sum (5.5) to be non-zero. Hence D(0) = D(1) = D(2) = 0. This fact is not surprising [7,13] since we indicated before that the geometric multiplicity of the eigenvalue λ = 0 is equal to 3. From (5.6), it is also easy to combine the non-zero terms in the expression for D(3) given by (5.5) to obtain (1)
(1)
(1)
∗(1)
(1)
∗(1)
D(3) = det(u1 − u4 , u2 − u5 (1)
∗(1)
(1)
∗(1)
, u3 − u6
(0)
(0)
(0)
, u1 , u2 , u3 ).
From (5.3), u2 = u5 and u3 = u6 , and D(3) is therefore equal to zero. As for the fourth term in the expansion (5.4), non-zero contributions to D(4) containing only terms of order zero and one add up to (1)
(0)
(0)
(1)
∗(1)
− u2 , u6
(1)
∗(1)
− u3 ) + det(u1 , u2 , u3 , u4 − u1 , u5
(0)
(0)
(1)
(1)
(1)
∗(1)
− u2 , u6
det(u1 , u2 , u3 , u4 , u5
+ det(u1 , u2 , u3 , u4 − u1 , u5
(1)
(1)
(0)
∗(1)
(1)
(0)
(1)
(1)
(2)
(2)
(2)
(1)
∗(1)
(1)
∗(1)
, u3 − u6
(0)
(1)
(2)
∗(2)
, u3 − u6
(1)
(1)
(1)
∗(1)
, u3 − u6
+ det(u1 − u4 , u2 − u5
(1)
− u3 )
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
is given by
, u1 , u2 , u3 )
(1)
+ det(u1 − u4 , u2 − u5
∗(1)
, u6
),
which is zero. The contribution to D(4) coming from non-zero terms containing any ui det(u1 − u4 , u2 − u5
∗(1)
(1)
∗(1)
, u1 , u2 , u3 )
(2)
∗(2)
, u1 , u2 , u3 ),
which is also zero because of relation (5.3). Even though λ = 0 is not an isolated eigenvalue of L, the vanishing of D(3) and D(4) confirms that the algebraic multiplicity of λ = 0 is equal to 5 [7]. This can indeed be understood as follows. One can easily check that an equation of the form φ(0) = A(ξ, 0)φ(0) is equivalent to LY0 = 0, where Y0 = (φ(0) [1], φ(0) [2], φ(0) [3], 0, 0, 0)T and φ(0) [i] refers to the ith component of φ(0) (ξ). Similarly, the equation ∂A(ξ, λ) (1) (1) = A(ξ, 0)φ + φ(0) φ ∂λ λ=0 reads LY1 = Y0 with Y1 = (φ(1) [1], φ(1) [2], φ(1) [3], φ(0) [1], φ(0) [2], φ(0) [3])T and ∂A(ξ, λ) 1 ∂2 A(ξ, λ) (2) (2) (1) = A(ξ, 0)φ + φ + φ(0) φ ∂λ λ=0 2 ∂λ2 λ=0 reads LY2 = Y1 with Y2 = (φ(2) [1], φ(2) [2], φ(2) [3], φ(1) [1], φ(1) [2], φ(1) [3])T . As a consequence, for each bounded φ(0) , the existence of a solution φ(1) bounded at +∞ and −∞ is equivalent to increasing by 1 the algebraic multiplicity of the eigenvalue λ = 0. If both φ(1) and φ(2) are bounded, the algebraic multiplicity of λ = 0 is (1) (1) (0) increased by 2. Here, φ2 , which converges at +∞, and φ5 , which converges at −∞, are such that φ2 (ξ) = (0) (1) (1) φ5 (ξ) = X2 (ξ). The first relation in (5.3) indicates that φ2 and φ5 , corrected by linear combinations of the Xi ’s, i = 1, 2, 3, match at ξ = 0, which implies that there exists a bounded φ(1) for φ(0) = X2 . Similarly, the second relation in (5.3) implies that there is a bounded φ(1) for φ(0) = X3 . A consequence of (5.3) is therefore that the algebraic multiplicity of λ = 0 is equal to 5. In terms of the derivatives of D(λ) at λ = 0, Eq. (5.3) imply that D(3) and D(4) both vanish and that λ = 0 is a zero of the Evans function of order 5. As in the standard case [7], the order of the zero of the Evans function at λ = 0 is therefore equal to the algebraic multiplicity of this eigenvalue. In the
118
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
discussion of Section 3.1, we identified the two generalized eigenfunctions as the derivatives of the pulse solution with respect to ω and to c, respectively, taken at ω = 0. The first non-zero term in the expansion (5.4) is the sixth one. One can check that the terms in D(5) containing (2) at most one ui add up to zero and that D(5) is given by (1)
(1)
(2)
∗(2)
D(5) = det(u1 − u4 , u2 − u5
(2)
∗(2)
, u3 − u6
(0)
(0)
(0)
, u1 , u2 , u3 ).
Now (2)
∗(2)
u2 − u5
(2)
(2)
= u2 − u5 +
3 i=2
−
3 i=2
(1)
d2,i ui
(1) c5,i φi,part (0) −
(2)
= φ2,part (0) +
i=2
3 i=2
(0) e5,i ui
(1)
+ (c5,3 − c2,3 ) φ3,part (0) + (2)
(2)
3 i=2
(1)
c2,i φi,part (0) +
+ (c5,2 − c2,2 )
3 i=2
= φ2,part (0) − φ5,part (0) +
3
3 i=2
(1) φ2,part (0) +
3 i=2
(0)
(2)
e2,i ui − φ5,part (0) (0) c2,i ui
(0)
c3,i ui
(0)
(e2,i − e5,i + (c5,2 − c2,2 )c2,i + (c5,3 − c2,3 )c3,i )ui ,
and, similarly, (2)
∗(2)
u3 − u6
(2)
(2)
= u3 − u6 +
3 i=1
(2)
(1)
d3,i ui
(2)
= φ3,part (0) − φ6,part (0) + µ(c2
− 2c02 ) −4 βc3 (c2 − c02 )
3 i=2
(0)
(e3,i − e6,i + (c6,2 − c3,2 )c2,i + (c6,3 − c3,3 )c3,i )ui
3 X4 (0) (0) . + c1,i ui 2µ(c − 1) i=1
As a consequence (1)
(1)
(2)
(2)
(2)
(2)
(0)
(0)
(0)
D(5) = det(u1 − u4 , φ2,part (0) − φ5,part (0), φ3,part (0) − φ6,part (0), u1 , u2 , u3 ) µ(c2 − 2c02 ) X4 (0) (1) (1) (2) (2) (0) (0) (0) + det u1 − u4 , φ2,part (0) − φ5,part (0), −4 , u , u2 , u3 . βc3 (c2 − c02 ) 2µ(c − 1) 1 Since (1)
(1)
u1 − u4 =
X4 (0) X4 (0) (0) − + (c1,i − c4,i )ui , 2µ(c − 1) 2µ(c + 1) 3
i=1
the second determinant in this expression is zero and an evaluation of the first determinant gives D(5) =
128 c02 µ2 (3c02 (3c2 − 2c02 ) − c4 (c02 + 2)) . 3 c6 (c2 − c02 )4
(5.7)
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
119
5.3. Behavior of the Evans function for large λ In order to study the behavior of the Evans function D(λ) as λ → ∞, we follow [13] and introduce the change of variable ζ = λξ,
(5.8)
which does not change the sign of the Evans function for λ ∈ R, λ > 0. Then Eq. (3.2) becomes ˜ λ)X, X = A(ζ,
X = (u, v, w, u , v , w )T ,
where the prime now means derivative with respect to ζ and 0 0 0 0 0 0 0 0 0 2 a (ζ/λ) C1 − 0 0 2 2 2 2 ˜ λ) = λ (c − c0 ) λ (c2 − c02 ) A(ζ, a2 (ζ/λ) C1 − 0 0 λ2 (c2 − c2 ) 2 (c2 − c2 ) λ 0 0 a (ζ/λ) a0 (ζ/λ) 1 − 20 2 − 2 2 2 λ (c − 1) λ (c − 1) c −1 with
(5.9)
c2
1 0 0
0 1 0
0 0 1
2c − c02
0
a0 (ζ/λ) λ(c2 − c02 )
2c c2 − c02
a0 (ζ/λ) λ(c2 − c02 )
0 −
a0 (ζ/λ) λ(c2 − 1)
−
a0 (ζ/λ) λ(c2 − 1)
2c −1
,
c2
2 ζ ζ C1 = −λ + µ − 2 a0 + b0 . λ λ 2
Under the change of variable (5.8), the solutions φi to (3.2) defined in Section 3 give rise to solutions of system (5.9) ζ φ˜ i (ζ, λ) = φi ,λ , λ ˜ λ) and the Evans function D(λ) can be expressed in terms of the φ˜ i (0, λ). As λ → +∞, λ ∈ R+ , the matrix A(ζ, ˜ 0 . Because the system (5.9) has an exponential dichotomy for λ converges uniformly toward a constant matrix A ˜ λ)X are uniformly close to the solutions of the system X = A ˜ 0 X [13]. The sign of large, solutions of X = A(ζ, D(λ) for large positive λ can therefore be found by calculating the Evans function for this system, provided the ˜0 basis vectors are arranged in the same order. This can easily be accomplished by comparing the eigenvectors of A with the images, under the change of variable (5.8), of the eigenvectors defined in (3.14). More precisely, we have ∞ lim φ˜ 1 (ζ, λ) e−˜ν2 (λ)ζ = v˜ ∞ 2 (λ),
ζ→∞
∞ lim φ˜ 2 (ζ, λ) e−˜ν3 (λ)ζ = w ˜ 2 (λ),
ζ→∞
∞ lim φ˜ 3 (ζ, λ) e−˜ν3 (λ)ζ = w ˜ 3 (λ),
ζ→∞
(5.10) and ∞ lim φ˜ 4 (ζ, λ) e−˜ν1 (λ)ζ = v˜ ∞ 1 (λ),
ζ→−∞
∞ lim φ˜ 5 (ζ, λ) e−˜ν4 (λ)ζ = w ˜ 5 (λ),
ζ→−∞
∞ lim φ˜ 6 (ζ, λ) e−˜ν4 (λ)ζ = w ˜ 6 (λ),
ζ→−∞
(5.11)
120
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
where the eigenvalues ν˜ i∞ and the eigenvectors v˜ ∞ i are related to the eigenvalues (3.13) and eigenvectors (3.14) of A∞ (λ) by ν˜ i∞ (λ) =
νi∞ (λ) , λ
∞ (˜v∞ i (λ))j = (vi (λ))j ,
On the other hand, the matrix 0 0 0 0 0 0 1 0 ˜ 0 = − c2 − c2 A 0 1 0 − 2 c − c02 0 0
0 0 0
1 0 0 2c c2 − c02
0 0 −
c2
(˜v∞ i (λ))j =
j = 1, 2, 3,
0
1 −1
c2
0
0 1 0
0 0 1
0
0
2c − c02
0
0
2c −1
(v∞ i (λ))j , λ
j = 4, 5, 6.
c2
has eigenvalues ν˜ i0 ν˜ 10 =
1 , c+1
ν˜ 20 =
1 , c−1
ν˜ 30 =
1 , c − c0
ν˜ 40 =
1 , c + c0
and eigenvectors 0 0 1 0 v˜ 1 = 0 , 0 ν˜ 10
v˜ 04,1
0 0 1 0
0
2 (c − c02 )˜ν30 − 2c 0 0 0 v˜ 2 = v˜ 3,1 = , , 0 0 −1 ν˜ 20 0 0 (c2 − c02 )˜ν40 − 2c (c2 − c2 )˜ν0 − 2c 0 4 0 0 0 , 0 = v˜ 4,2 = , −1 0 0 −1 0 0
v˜ 03,2
=
(c2 − c02 )˜ν30 − 2c
where, as before, ν˜ 30 and ν˜ 40 have multiplicity two. It can easily be checked that, as λ → ∞ v˜ ∞ ˜ 0i , i (λ) ∝ v
i = 1, 2,
and
w ˜ i (λ) ∝ w ˜ 0i ,
i = 2, 3, 5, 6,
where the ∝ symbol indicates that the two vectors point in the same direction, and 2α (˜v0 − v˜ 03,2 ), β(c2 − c02 ) 3,1 2α w ˜ 05 = (−˜v04,1 + v˜ 04,2 ), 2 β(c − c02 ) w ˜ 02 =
2α (˜v0 + v˜ 03,2 ), c2 − c02 3,1 2α w ˜ 06 = (˜v0 + v˜ 04,2 ). 2 c − c02 4,1
w ˜ 03 =
0 0 −1 0 0
,
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
121
Therefore, as λ → ∞, the sign of the Evans function D(λ) is the same as the sign of ˜ = det[˜v02 , w D ˜ 02 , w ˜ 03 , v˜ 01 , w ˜ 05 , w ˜ 06 ] =
512c02 α4 µ(c2 − 1)(c2 − c02 )3
,
(5.12)
which is negative. 5.4. Instability criterion We have just shown that the Evans function D(λ) is negative for λ real and sufficiently large. Therefore, if the expression given in (5.7) happens to be positive, the Evans function must have at least one zero on the positive real line. The pulse solution (2.4) is thus unstable if T ≡ 3c02 (3c2 − 2c02 ) − c4 (c02 + 2) > 0. It is only a matter of calculations to prove that, together with conditions (2.6), T > 0 can be re-written as
2 9 − 33 − 24c02 c0 K≡ < c2 < c02 < 1. 2 c02 + 2
(5.13)
This defines a non-empty set of pulse velocities c since K < c02 when c02 < 1. In the case of Fig. 1, c 0.6124, c02 = 12/5 and T = −27.078 < 0, so our criterion does not indicate instability. In fact, the numerical calculation shows that in this case the pulse is stable. At the bifurcation, when T = 0, the algebraic multiplicity of λ = 0 is (2) − φ(2) increased by 1 and is then equal to 6. A look at the formulae given in the appendix show that φ6,part 3,part is a (2) linear combination of X1 , X2 , X3 and X4 when T = 0 (since the coefficient of X6 in φ6,part vanishes). Since the (2) − φ(2) difference φ4(1) − φ1(1) also involves X1 , X2 , X3 and X4 , one can find a linear combination of φ6,part 3,part and (1) (1) φ4 − φ1 which only depends on X1 , X2 and X3 , i.e. which is bounded at ±∞. In agreement with the discussion of Section 5.2, this implies that there exists a bounded Y such that L3 Y = 0.
6. Conclusions In this article, we have considered a one-parameter family of pulse solutions (2.4) to two coupled non-linear Klein–Gordon equations (2.1). Using symmetry properties of (2.1), we were able to find an expansion of the Evans function associated with the linearization of (2.1) about these pulse solutions, for small values of the spectral parameter. The first non-zero derivative of the Evans function at the origin was obtained in (5.7) and the sign of the Evans function for λ real sufficiently large was shown to be negative in (5.12). A parity argument was then used to show instability of the pulse when criterion (5.13) holds. The calculations performed in this paper were not algorithmic and several difficulties had to be overcome. First, we had to deal with a linear system (3.2) for which the spaces of solutions bounded at +∞ and −∞ were both of dimension 3 (see however [11,14,15,17,21], for examples with stable spaces of dimension 2). As far as we know, such a situation was never considered in the study of a specific example. Moreover, because the linearized system does not have an exponential dichotomy when λ = 0, singular expansions of the solutions in powers of the spectral parameter had to be sought. As mentioned above, such expansions are typically used near a branch point of the Evans function [12,19]. The loss of exponential dichotomy at λ = 0 corresponds to the vanishing of the spectral gap [11,12] of the stable and unstable subspaces of the asymptotic matrix and is for instance a standard phenomenon in the case of conservation laws [11]. Finally, characterizing the behavior of the Evans function near the origin was
122
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
not a simple task since the various symmetries of the coupled non-linear Klein–Gordon equations and of its family of pulse solutions implied that the Evans function, as well as its first four derivatives, vanish at the origin. Criterion (5.13) guarantees instability but it is very likely that some pulse solutions of (2.1) are also unstable when this condition is not satisfied. In particular, numerical simulations have shown the presence of a marginal instability related to the dilation invariance (see Eq. (3.8)), in which B grows linearly in space. Since perturbations associated with this instability are clearly not bounded on the real line, they are not considered in the present study. As for other interesting problems, we would like to point out that a set of equations generalizing (2.1) for rods with non-circular cross-section was derived in [26]. The method discussed here could also be extended to pulse solutions of these generalized equations.
Acknowledgements We are grateful to M. Agrotis, N. Ercolani, A. Espiñola-Rocha, A. Goriely and R. Indik for useful discussions. Most of the calculations presented in this paper would not have been possible without the help of MAPLE (Waterloo Maple Inc.). One of the authors, SL acknowledges a postdoctoral fellowship from NSERC (National Science and Engineering Research Council) of Canada. This material is based upon work supported by the National Science Foundation under Grant No. DMS-0075827 to JL.
Appendix A Using the method of variation of parameters, we first find the following particular solutions to the inhomogeneous (2) (2) equations satisfied by φ2 and φ3 (again, we only show the first three components of each vector):
α(c02 e2βξ + 2β2 c2 ξ 2 + 2ξβ(c2 + c02 ) + c2 − c02 )
4µ(c2 − c02 ) cosh (βξ) (2) φˆ 2,part (ξ) = α(c02 e2βξ + 2β2 c2 ξ 2 + 2ξβ(c2 + c02 ) + c2 − c02 ) − 4µ(c2 − c02 ) cosh (βξ)
,
0 α(γ1 e−βξ + γ2 eβξ + γ3 e−3βξ ) 48βc2 (c2 − c02 )3 (c2 − 1)2 cosh 2 (βξ)
α(γ1 e−βξ + γ2 eβξ + γ3 e−3βξ ) ˆφ(2) (ξ) = 3,part 48βc2 (c2 − c02 )3 (c2 − 1)2 cosh 2 (βξ) δ1 cosh 2 (βξ) ln(1 + e−2βξ ) + δ2 e−2βξ + δ3 + δ4 e2βξ 6c4 (c2 − c02 )2 (c2 − 1)2 cosh 2 (βξ)
,
where γ1 = 3c2 (c2 − 1)(4c2 µ(c2 − 1)ξ 2 + 4β(c2 − 5)(c2 − c02 )2 ξ − (c2 − c02 )(5c4 − 7c2 + 4c2 c02 − 2c02 )), γ2 = −12c4 µ(c2 − 1)2 ξ 2 − 12c2 β(c2 − 1)2 (c2 − c02 )2 ξ +3(c2 − c02 )(5c8 − 16c6 − 2c6 c02 − 2c4 c02 + 27c4 + 12c2 c04 − 28c2 c02 + 4c04 ),
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
123
γ3 = 2(c2 − 1)(c2 − c02 )(c4 c02 + 2c4 − 9c2 c02 + 6c04 ), δ1 = 4(c2 − 1)(c2 − c02 )(6c04 − c4 + 7c4 c02 − 3c2 c02 − 9c2 c04 ), δ2 = 3c2 (c2 − 1)2 (c2 − c02 )(2βξ(c2 − 2c02 ) − c02 ), δ3 = −6µc4 (c2 − 1)2 ξ 2 − 6βc2 (c2 − 1)(c2 − c02 )(c2 c02 − 2c2 + c02 )ξ + 21 (c2 − c02 )(9c8 − 16c6 + 10c6 c02 − 70c4 c02 + 31c4 + 12c2 c02 + 48c2 c04 − 24c04 ), δ4 = 3(c2 − 1)(c2 − c02 )(2c02 − c2 )(c4 − 3c2 + 2c02 ). Particular solutions which converge as ξ → ∞ are given by (2)
(2)
φ2,part (ξ)=φˆ 2,part (ξ) − 2
c02 β (c2 − c02 )2
X5 (ξ),
(2)
(2)
φ3,part (ξ) = φˆ 3,part (ξ)−2 (2)
(2c02 − c2 )(c4 − 3c2 + 2c02 ) c4 (c2 − c02 )(c2 − 1)
X1 (ξ).
(2)
Particular solutions to the inhomogeneous equations satisfied by φ5 and φ6 , which converge as ξ → −∞, are given by (2) (2) φ5,part (ξ) = φˆ 2,part (ξ), (2)
(2)
φ6,part (ξ) = φˆ 3,part (ξ) + +
c02 (2µc2 − α2 c2 − 2µ) µc2 (c2 − 1)(c2 − c02 )
X1 (ξ) +
2 c4 c02 − c4 + 3c2 c02 − 3c04 X4 (ξ) 3 βc2 (c2 − 1)(c2 − c02 )2
8 c4 c02 + 2c4 − 9c2 c02 + 6c04 X6 (ξ). 3 βc2 (c2 − 1)(c2 − c02 )2
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
A. Goriely, M. Tabor, Nonlinear dynamics of filaments. II. Nonlinear analysis, Physica D 105 (1997) 45–61. J. Lega, A. Goriely, Pulses, fronts and oscillations of an elastic rod, Physica D 132 (1999) 373–391. G. Dangelmayr, E. Knobloch, The Takens–Bogdanov bifurcation with O(2)-symmetry, Phil. Trans. R. Soc. Lond. A 322 (1987) 243–279. J.W. Evans, Nerve axon equations. IV. The stable and unstable impulse, Ind. Univ. Math. J. 24 (1975) 1169–1190. C.K.R.T. Jones, Stability of the travelling wave solution to the FitzHugh–Nagumo equation, Trans. AMS 286 (1984) 431–469. E. Yanagida, Stability of fast traveling pulse solutions of the FitzHugh–Nagumo equations, J. Math. Biol. 22 (1985) 81–104. J. Alexander, R. Gardner, C. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410 (1990) 167–212. R. Pego, M. Weinstein, Eigenvalues, and instabilities of solitary waves, Phil. Trans. R. Soc. Lond. A 340 (1992) 47–94. J. Swinton, The stability of homoclinic pulses: a generalisation of Evans’s method, Phys. Lett. A 163 (1992) 57–62. T.J. Brigdes, G. Derks, Hodge duality and the Evans function, Phys. Lett. A 251 (1999) 363–372. R.A. Gardner, K. Zumbrun, The Gap Lemma and geometric criteria for instability of viscous shock profiles, Commun. Pure Appl. Math. LI (1998) 0797–0855. T. Kapitula, B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Physica D 124 (1998) 58–103. B. Sandstede, Stability of travelling waves, in: B. Fiedler (Ed.), Handbook of Dynamical Systems. II. Towards Applications, Elsevier, Amsterdam, 2002, pp. 983–1055. J.C. Alexander, C.K.R.T. Jones, Existence and stability of asymptotically oscillatory double pulses, J. Reine Angew. Math. 446 (1994) 49–79. J.C. Alexander, C.K.R.T. Jones, Existence and stability of asymptotically oscillatory triple pulses, Z. Angew. Math. Phys. 44 (1993) 189–200. T. Kapitula, The Evans function and generalized Melnikov integrals, SIAM J. Math. Anal. 30 (1998) 273–297. T. Kapitula, Stability criterion for bright solitary waves of the perturbed cubic–quintic Schrödinger equation, Physica D 116 (1998) 95–120. T. Kapitula, B. Sandstede, Edge bifurcations for near integrable systems via Evans function techniques, SIAM J. Math. Anal. 33 (2002) 1117–1143. T. Kapitula, J. Rubin, Existence and stability of standing hole solutions to complex Ginzburg–Landau equations, Nonlinearity 13 (2000) 77–112.
124
S. Lafortune, J. Lega / Physica D 182 (2003) 103–124
[20] T.J. Bridges, G. Derks, Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry, Proc. R. Soc. Lond. A 455 (1999) 2427–2469. [21] T.J. Bridges, G. Derks, The symplectic Evans matrix, and the instability of solitary waves and fronts, Arch. Rat. Mech. Anal. 156 (2001) 1–87. [22] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry: II, J. Funct. Anal. 94 (1990) 308–348. [23] P. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1993. [24] E. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955, p. 104. [25] Y.A. Li, K. Promislow, The mechanism of the polarizational mode instability in birefringent fiber optics, SIAM J. Math. Anal. 31 (2000) 1351–1373. [26] A. Goriely, M. Nizette, M. Tabor, On the dynamics of elastic strips, J. Nonlinear Sci. 11 (2001) 3–45.